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RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE Exponents and Polynomials Section 7A Section 7B Exponents Polynomials 7-1 Integer Exponents 7-6 Polynomials 7-2 Powers of 10 and Scientific Notation 7-7 Algebra Lab Model Polynomial Addition and Subtraction 7-3 Algebra Lab Explore Properties of Exponents 7-7 Adding and Subtracting Polynomials 7-3 Multiplication Properties of Exponents 7-8 Algebra Lab Model Polynomial Multiplication 7-4 Division Properties of Exponents 7-8 Multiplying Polynomials 7-5 Rational Exponents Connecting Algebra to Geometry Volume and Surface Area 7-9 Special Products of Binomials Pacing Guide for 45-Minute Classes Calendar Planner® Chapter 7 Countdown Weeks 15 , 16 , 17 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 7-1 Lesson 7-2 Lesson 7-3 Algebra Lab 7-3 Lesson 7-4 Lesson DAY 6 DAY 7 DAY 8 DAY 9 DAY 10 7-5 Lesson Multi-Step Test Prep Ready to Go On? 7-6 Lesson 7-7 Algebra Lab 7-7 Lesson 7-7 Lesson 7-8 Algebra Lab DAY 11 DAY 12 DAY 13 DAY 14 DAY 15 7-8 Lesson Connecting Algebra to Geometry 7-9 Lesson Multi-Step Test Prep Ready to Go On? Chapter 7 Review DAY 16 Chapter 7 Test Pacing Guide for 90-Minute Classes Calendar Planner® Chapter 7 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 Chapter 6 Test 7-1 Lesson 7-2 Lesson 7-3 Algebra Lab 7-3 Lesson 7-4 Lesson 7-5 Lesson Multi-Step Test Prep Ready to Go On? 7-6 Lesson 7-7 Algebra Lab 7-7 Lesson DAY 6 DAY 7 DAY 8 DAY 9 7-7 Lesson 7-8 Algebra Lab 7-8 Lesson Connecting Algebra to Geometry 7-9 Lesson Multi-Step Test Prep Ready to Go On? Chapter 7 Review Chapter 7 Test 8-1 Lesson 456A Chapter 7 E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS DIAGNOSE Assess Prior Knowledge PRESCRIBE Before Chapter 7 Diagnose readiness for the chapter. Prescribe intervention. Are You Ready? SE p. 457 Are You Ready? Intervention Before Every Lesson Diagnose readiness for the lesson. Prescribe intervention. Warm Up TE Reteach CRB During Every Lesson Diagnose understanding of lesson concepts. Prescribe intervention. Check It Out! SE Questioning Strategies TE Think and Discuss SE Write About It SE Journal TE Reading Strategies CRB Success for Every Learner Lesson Tutorial Videos After Every Lesson Formative Assessment Diagnose mastery of lesson concepts. Prescribe intervention. Lesson Quiz TE Test Prep SE Test and Practice Generator Reteach CRB Test Prep Doctor TE Homework Help Online Before Chapter 7 Testing Diagnose mastery of concepts in chapter. Prescribe intervention. Ready to Go On? SE pp. 495, 529 Multi-Step Test Prep SE pp. 494, 528 Section Quizzes AR Test and Practice Generator Ready to Go On? Intervention Scaffolding Questions TE pp. 494, 528 Reteach CRB Lesson Tutorial Videos Before High Stakes Testing Diagnose mastery of benchmark concepts. Prescribe intervention. College Entrance Exam Practice SE p. 535 Standardized Test Prep SE pp. 538–539 College Entrance Exam Practice After Chapter 7 Summative Assessment KEY: SE = Student Edition Check mastery of chapter concepts. Prescribe intervention. Multiple-Choice Tests (Forms A, B, C) Free-Response Tests (Forms A, B, C) Performance Assessment AR Cumulative Test AR Test and Practice Generator Reteach CRB Lesson Tutorial Videos TE = Teacher’s Edition CRB = Chapter Resource Book AR = Assessment Resources Available online Available on CD- or DVD-ROM 456B RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE E Lesson Resources Before the Lesson Practice the Lesson Prepare Teacher One Stop JG8E@J? Practice Chapter Resources • Editable lesson plans • Calendar Planner • Easy access to all chapter resources • Practice A, B, C Practice and Problem Solving Workbook JG8E@J? IDEA Works!® Modified Worksheets and Tests ExamView Test and Practice Generator Lesson Transparencies • Teacher Tools JG8E@J? Homework Help Online JG8E@J? Online Interactivities Interactive Online Edition Teach the Lesson • Homework Help Introduce Alternate Openers: Explorations Lesson Transparencies Apply Chapter Resources • Warm Up • Problem of the Day • Problem Solving JG8E@J? Practice and Problem Solving Workbook JG8E@J? Interactive Online Edition Teach Lesson Transparencies • Chapter Project • Teaching Transparencies Project Teacher Support Know-It Notebook™ • Vocabulary • Key Concepts Power Presentations Lesson Tutorial Videos Interactive Online Edition After the Lesson JG8E@J? Reteach Chapter Resources • Reteach • Reading Strategies %,, • Lesson Activities • Lesson Tutorial Videos Lab Activities Lab Resources Online Online Interactivities TechKeys Success for Every Learner Review Interactive Answers and Solutions Solutions Key Know-It Notebook™ JG8E@J? • Big Ideas • Chapter Review Extend Chapter Resources • Challenge Technology Highlights for the Teacher Power Presentations Teacher One Stop Dynamic presentations to engage students. Complete PowerPoint® presentations for every lesson in Chapter 7. KEY: SE = Student Edition 456C Chapter 7 TE = Teacher’s Edition %,, JG8E@J? Easy access to Chapter 7 resources and assessments. Includes lesson planning, test generation, and puzzle creation software. English Language Learners JG8E@J? Spanish version available Premier Online Edition JG8E@J? Chapter 7 includes Tutorial Videos, Lesson Activities, Lesson Quizzes, Homework Help, and Chapter Project. Available online Available on CD- or DVD-ROM E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS Reaching All Learners Teaching tips to help all learners appear throughout the chapter. A few that target specific students are included in the lists below. All Learners On-Level Learners Lab Activities Practice and Problem Solving Workbook JG8E@J? Know-It Notebook Practice B ............................................................................. CRB Problem Solving .................................................................. CRB Vocabulary Connections ..............................................SE p. 458 Questioning Strategies ...........................................................TE Ready to Go On? Intervention JG8E@J? Know-It Notebook Homework Help Online JG8E@J? Online Interactivities Special Needs Students Practice A ............................................................................. CRB Reteach................................................................................. CRB Reading Strategies ............................................................... CRB Are You Ready?............................................................SE p. 457 Inclusion .....................................................TE pp. 461, 505, 513 IDEA Works!® Modified Worksheets and Tests Ready to Go On? Intervention JG8E@J? Know-It Notebook JG8E@J? Online Interactivities JG8E@J? Lesson Tutorial Videos Advanced Learners Practice C ............................................................................. CRB Challenge ............................................................................. CRB Challenge Exercises ...............................................................SE Reading and Writing Math Extend ..............................TE p. 459 Critical Thinking ..........................................................TE p. 515 Are You Ready? Enrichment JG8E@J? Developing Learners Practice A ............................................................................. CRB Reteach................................................................................. CRB Reading Strategies ............................................................... CRB Are You Ready?............................................................SE p. 457 Vocabulary Connections ..............................................SE p. 458 Questioning Strategies ...........................................................TE Ready to Go On? Intervention JG8E@J? Know-It Notebook Homework Help Online JG8E@J? Online Interactivities JG8E@J? Lesson Tutorial Videos Ready To Go On? Enrichment JG8E@J? English Language Learners ENGLISH LANGUAGE LEARNERS Reading Strategies ............................................................... CRB Are You Ready? Vocabulary ........................................SE p. 457 Vocabulary Connections ..............................................SE p. 458 Vocabulary Review ......................................................SE p. 530 English Language Learners................................TE pp. 459, 476 Success for Every Learner Know-It Notebook Multilingual Glossary JG8E@J? Lesson Tutorial Videos Technology Highlights for Reaching All Learners Lesson Tutorial Videos JG8E@J? Starring Holt authors Ed Burger and Freddie Renfro! Live tutorials to support every lesson in Chapter 7. KEY: SE = Student Edition TE = Teacher’s Edition Multilingual Glossary Searchable glossary includes definitions in English, Spanish, Vietnamese, Chinese, Hmong, Korean, and 4 other languages. CRB = Chapter Resource Book Online Interactivities Interactive tutorials provide visually engaging alternative opportunities to learn concepts and master skills. JG8E@J? Spanish version available Available online Available on CD- or DVD-ROM 456D RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE E Ongoing Assessment Assessing Prior Knowledge Lesson Assessment Determine whether students have the prerequisite concepts and skills for success in Chapter 7. Provide formative assessment for each lesson of Chapter 7. Are You Ready? JG8E@J? ............................SE p. 457 Warm Up .................................................................TE Test Preparation Provide review and practice for Chapter 7 and standardized tests. Multi-Step Test Prep ..........................................SE pp. 494, 528 Study Guide: Review .........................................SE pp. 530–533 Test Tackler ........................................................SE pp. 536–537 Standardized Test Prep......................................SE pp. 538–539 College Entrance Exam Practice ..................................SE p. 535 Countdown to Testing ........................ SE pp. C4–C27 ® IDEA Works! Modified Worksheets and Tests Questioning Strategies ...........................................................TE Think and Discuss ...................................................................SE Check It Out! Exercises ...........................................................SE Write About It .........................................................................SE Journal ....................................................................................TE Lesson Quiz .............................................................TE Alternative Assessment ..........................................................TE IDEA Works!® Modified Worksheets and Tests Weekly Assessment Provide formative assessment for each section of Chapter 7. Multi-Step Test Prep ..........................................SE pp. 494, 528 JG8E@J? .................SE pp. 495, 529 Ready to Go On? Section Quizzes JG8E@J? ........................................................ AR Test and Practice Generator JG8E@J? .... Teacher One Stop Alternative Assessment Assess students’ understanding of Chapter 7 concepts and combined problem-solving skills. Chapter Assessment Chapter 7 Project .........................................................SE p. 456 Alternative Assessment ..........................................................TE Performance Assessment JG8E@J? ........................................ AR Portfolio Assessment JG8E@J? ............................................... AR Provide summative assessment of Chapter 7 mastery. Chapter 7 Test ..............................................................SE p. 534 Chapter Test (Levels A, B, C) JG8E@J? ................................... AR • Multiple Choice • Free Response Cumulative Test JG8E@J? ....................................................... AR Test and Practice Generator JG8E@J? .... Teacher One Stop ® IDEA Works! Modified Worksheets and Tests Technology Highlights for Assessment Are You Ready? Ready to Go On? JG8E@J? Automatically assess readiness and prescribe intervention for Chapter 7 prerequisite skills. KEY: 456E SE = Student Edition Chapter 7 TE = Teacher’s Edition Automatically assess understanding of and prescribe intervention for Sections 7A and 7B. AR = Assessment Resources JG8E@J? Spanish version available Test and Practice Generator JG8E@J? Use Chapter 7 problem banks to create assessments and worksheets to print out or deliver online. Includes dynamic problems. Available online Available on CD- or DVD-ROM E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS Formal Assessment Three levels (A, B, C) of multiple-choice and free-response chapter tests, along with a performance assessment, are available in the Assessment Resources. A Chapter 7 Test A Chapter 7 Test C Chapter 7 Test C Chapter 7 Test MULTIPLE CHOICE FREE RESPONSE PERFORMANCE ASSESSMENT B Chapter 7 Test B Chapter 7 Test Chapter 7 Test Select the best answer. 9. Simplify (3a5)2. 1. Which of the following is equivalent to 2−3? A (−2)(−2)(−2) B − C 1 (2)(2)(2) 1 (2)(2)(2) D (2)(2)(2) 2. Evaluate (3 + x)−2 for x = −1. F −4 G H 1 4 10 9 1 4b3 1 64b3 D 4 b3 F 2.67 × 106 H 26.7 × 104 7 5 F 10−5 H 104 G 10−4 J 105 5. In 2000, the population of Texas was about 20,900,000 people. Write this population in scientific notation. A 2.09 × 105 C 20.9 × 106 B 2.09 × 107 D 209 × 105 H 0.062 1 620 J 6200 a . a2 ________________________________________ A a4 C a10 B a6 D a16 4. Write 0.0000001 as a power of 10. 20 F 2y x3 H G 2y 20 x15 J 32y5 32y x15 5. In 2000, the population of Indiana was about 6,080,000 people. Write this population in scientific notation. −2 11. . ________________________________________ ⎛ 4y 6 ⎞ 12. ⎜ 3 ⎟ ⎝ x ⎠ 6. Write 8.5 × 10−7 in standard form. b8 25 A 1 25b 4 C B 5 b6 D −5 ________________________________________ 2 2 ________________________________________ ⎛ 0⎞ 13. ⎜ 3 ⎟ ⎝b ⎠ H x9 J x18 7. y7 • y3 ________________________________________ ________________________________________ A 4 2 A 2x + 2x − 4 C 6 B 5 B 9x2 − 5x − 4 D 64 14. Simplify the quotient (6.4 × 109) ÷ (1.6 × 10−3). Write the answer in scientific notation. C 9x2 − 2x − 7 2 ________________________________________ 4 F 2 H 16 G 8 22. Subtract (7a2 − 3a) − (5a2 − 5a). J 512 3 2 C x B x3 Simplify. D 9x − 5x − 4 17. Simplify (x 2 )4 x 3 . All variables represent nonnegative numbers. 2(x) D x9 18. When written in standard form, which polynomial has a leading coefficient of 5? F −7 + 6y + 5y2 G x+5 H x2 − 5x3 + 2x 2 J 5y + 3y − 4 19. Classify the polynomial 3x5 + 3 according to its degree and number of terms. A cubic binomial B cubic trinomial F 2a2 − 8a H 4 G 2a2 + 2a J 12a2 − 8a F 3w2 − 2 H 4w − 2 G 3w2 − 2w J 8w − 4 25. Multiply (x − 5)(2x + 4). C 2x2 − 20 A −6x 2 2 B 2x − 26 D 2x − 6x − 20 G b3 − 2b2 − 22b − 21 J 464 ft2 ________________________________________ 17. Simplify (x 3 )3 5 y 5 . All variables D quintic trinomial G 336 ft2 Converts between standard notation and engineering notation. ______ Compares and contrasts scientific and engineering notation. Scoring Rubric Level 1: Student is not able to solve any of the problems. Chapter 7 Test (continued) 21. Add (5x2 − 2x + 9) + (2x 2 − 4). ________________________________________ 22. Subtract (10a2 − 6a) − (7a2 − 8a). 23. Multiply (−5rs4)(3r 5s2). 24. A rectangle has width w and its length is 2 units shorter than 3 times the width, or 3w − 2. Write a polynomial for the area of the rectangle. F b3 − 5b2 − 21 H 384 ft2 Uses powers of 10 to represent periods. ______ Commas are used to separate a large number into groups of three digits, or periods. Decimal numbers less than 1 can be written with spaces to distinguish periods. Periods provide a shorthand way to write the numbers. 360 , millions period 000 , thousands period 000 = 360 million H 3b3 − 15b2 − 21b 2 J 4b − 20b − 28b 2 27. Multiply (2x + 7) . A 2x2 + 7 C 4x2 + 14x + 49 B 4x2 + 49 D 4x2 + 28x + 49 28. Which product results in x2 − 100? F (x − 10)2 G (x + 10)2 H x(x − 100) 0. 000 , units period 000 , thousandths period millionsths period 045 = 45 billionths billionths period Write these numbers using periods (trillion, billion, million, thousand, thousandths, millionths, billionths, and trillionths). 1. 35,000,000,000,000 26. Multiply (b + 3)(b2 − 5b − 7). F 114 ft2 1 15. 32 5 16. 3 Rewrites numbers using words to describe periods. ______ ________________________________________ 23. Multiply (4rs3)(6r 2s3). A 10r 2s9 C 24r 2s9 B 10r 3s6 D 24r 3s6 C quintic binomial 20. Brett has 100 feet of fence with which to make a rectangular cage for his dog. The area of the cage in square feet is given by the polynomial −w2 + 50w, where w is the width of the cage in feet. What is the area of the cage if the width is 8 feet? As a class, go over the two introductory examples. Mention that the use of the comma, space, and decimal point to separate periods is a regional, North American convention. Around the world, there are a variety of symbols used to denote separators in multi−digit numbers. Level 3: Student solves most problems correctly but gives a faulty comparison between scientific and engineering notation. B Chapter 7 Test (continued) 21. Add (2x2 − 5x − 7) + (7x 2 + 3). 16. Simplify 64 3 . Many scientific and graphing calculators have both a scientific notation mode (SCI) and an engineering notation mode (ENG). If you routinely use calculators in your class, you may want to allow students to use them on this activity. Calculators can be particularly beneficial for problem 16 because students can enter and compare numbers under both notations. Level 2: Student solves a few problems and gives no comparison. B Chapter 7 Test (continued) 1 Overview Level 4: Student solves all problems correctly and gives a solid comparison between scientific and engineering notation. ________________________________________ 15. Simplify 256 4 . Preparation Hints ______ Simplify. J 4.8 × 10 F x2 Individuals Performance Indicators −2 −24 G x3 Grouping Introduce the Task 8. (x20)4 H 3 × 10−3 30−40 minutes Students first rewrite numbers using periods. Then they write the periods using powers of 10, which have exponents that are always multiples of 3. Students then use engineering notation, a variation of scientific notation, to rewrite numbers. a16 a4 ________________________________________ G 3 × 10−24 D y50 8. Simplify (x ) . A x Simplify. ________________________________________ 20 Time Review powers of 10 and scientific notation. Review the place-value names of digits in decimal numbers and define groups of digits as periods. 3. Simplify 5b−8. F 1.728 × 10−12 6 3 1 10. A computer modem can transmit 1.5 × 106 bytes per second. How many bytes can it transmit in 300 seconds? Write your answer in scientific notation. ________________________________________ 14. Simplify the quotient (7.2 × 10−18) ÷ (2.4 × 106). C y15 B y5 ________________________________________ 8 ⎛ 5⎞ 13. Simplify ⎜ 4 ⎟ ⎝b ⎠ 7. Simplify y10 • y5. A y2 ________________________________________ ⎛ 2y 4 ⎞ 12. Simplify ⎜ 3 ⎟ . ⎝ x ⎠ 6. Which of the following is the standard form of 6.2 × 10−2? F −620 J 80.1 × 10 This performance task assesses the student’s ability to use powers of 10 to convert between standard, period, engineering, and scientific notations. ________________________________________ 2. Evaluate (5 − x)−2 for x = −1. Purpose 9. (2a3)5 5 4. Which power of 10 is equivalent to 0.00001? G D 9a10 G 8.01 × 10 3. Simplify 4b−3. B C 9a5 B 3a10 11. Simplify C 1. Simplify 4 . A 3a7 10. In June 2005, Miami International Airport served about 8.9 × 104 airline passengers per day. Find the approximate number of airline passengers served in total during the 30 days of June. J 10 A −64b3 −3 4 27 3 ________________________________________ ________________________________________ 24. A rectangle has width w and its length is 4 units longer than 2 times the width, or 2w + 4. Write a polynomial for the area of the rectangle. 1 represent nonnegative numbers. ________________________________________ 18. When the polynomial 5x − 2x3 + 8x2 − 7 is written in standard form, what is the leading coefficient? ________________________________________ 20. Genie has 100 feet of fence with which to make a rectangular cage for her rabbit. If she uses the wall of her house as one side, the area of the cage in square feet is given by the polynomial −2w2 + 100w, where w is the width of the cage in feet. What is the area of the cage if the width is 15 feet? 4. 0.000 012 ________________________ ________________________________________ Multiply. 25. (x + 6)(3x − 8) ________________________ 10. 1 thousand ________________________ ________________________________________ 2. 425,000 3. 9,500,000 ________________________ 5. 0.000 000 000 005 ________________________ ________________________ 6. 0.069 ________________________ Write these numbers as powers of 10. 7. 1 trillion 8. 1 billion 9. 1 million ________________________ 11. 1 thousandth ________________________ 12. 1 millionth ________________________ ________________________ Engineering notation is a method of writing a number with a power of 10 to emphasize the period of the number. It helps to write the period first. 2 26. (b − 4)(b + 3b − 2) ________________________________________ 19. Classify the polynomial 5x2 + 9x + 1 according to its degree and number of terms. ________________________ ________________________________________ 27. (3x − 4)2 ________________________________________ 28. (2x + 4)(2x − 4) ________________________________________ 0.000 000 045 = 45 billionths = 45 109 Write these numbers in engineering notation. 13. 96,000,000,000 ________________________ 14. 108,000 15. 0.000 004 ________________________ 16. Based on your answers to problems 715, complete this table to compare and contrast scientific notation and engineering notation. Notation ________________________ Leading factor Power of 10 Scientific Engineering 17. Convert 5.8 105 to engineering notation. ______________________________ ________________________________________ J (x + 10)(x − 10) JG8E@J? Test & Practice Generator Modified chapter tests that address special learning needs are available in IDEA Works!® Modified Worksheets and Tests. Create and customize Chapter 7 Tests. Instantly generate multiple test versions, answer keys, and Spanish versions of test items. 456F Exponents and Polynomials SECTION 7A Exponents 7A 7-1 Exponents On page 494, students write, solve, and graph equations to model real-world speed-of-light situations. Exercises designed to prepare students for success on the Multi-Step Test Prep can be found in each lesson. SECTION 7B 7-2 Powers of 10 and Scientific Notation Lab Explore Properties of Exponents 7-3 Multiplication Properties of Exponents 7-4 Division Properties of Exponents 7-5 Rational Exponents 7B Polynomials 7-6 Polynomials Lab Model Polynomial Addition and Subtraction Polynomials On page 528, students multiply polynomials to model a real-world Integer Exponents 7-7 Adding and Subtracting Polynomials Lab Model Polynomial Multiplication 7-8 Multiplying Polynomials 7-9 Special Products of Binomials area situation. Exercises designed to prepare students for success on the Multi-Step Test Prep can be found in each lesson. • Use exponents and scientific notation to describe numbers. • Use laws of exponents to simplify monomials. • Perform operations with polynomials. Interactivities Online ▼ Every Second Counts How many seconds until you graduate? The concepts in this chapter will help you find and use large numbers such as this one. KEYWORD: MA7 ChProj Lessons 7-2, 7-5, 7-7 Lesson Tutorials Online 456 Chapter 7 Every Second Counts About the Project Project Resources In the Chapter Project, students consider the passage of time in seconds. First they calculate large numbers of seconds, such as how many seconds they’ve been alive or how many seconds until graduation. Then they calculate fractions of seconds as they learn about a car’s braking distance and driver reaction time. In each case, students use exponents and scientific notation to work with these very large and very small numbers. All project resources for teachers and students are provided online. A11NLS_c07_0456-0459.indd 456 Lesson Tutorial Videos are available for EVERY example. 456 Chapter 7 Materials: • calculators KEYWORD: MA7 ProjectTS 8/18/09 9:50:17 AM Vocabulary Match each term on the left with a definition on the right. 1. Associative Property F A. a number that is raised to a power Organizer 2. coefficient B B. a number that is multiplied by a variable 3. Commutative Property C C. a property of addition and multiplication that states you can add or multiply numbers in any order 4. exponent D D. the number of times a base is used as a factor 5. like terms E E. terms that contain the same variables raised to the same powers Objective: Assess students’ understanding of prerequisite skills. F. a property of addition and multiplication that states you can group the numbers in any order Assessing Prior Knowledge Exponents INTERVENTION Write each expression using a base and an exponent. 6. 4 · 4 · 4 · 4 · 4 · 4 · 4 4 7 7. 5 · 5 5 2 9. x · x · x x 3 8. (-10)(-10)(-10)(-10) (-10)4 10. k · k · k · k · k k 5 11. 9 9 1 13. -12 2 -144 14. 5 3 125 Evaluate Powers Evaluate each expression. 12. 3 4 81 5 16. 4 64 17. (-1) 1 19. 25.25 × 100 2525 20. 2.4 × 6.5 15.6 Use this page to determine whether intervention is necessary or whether enrichment is appropriate. Resources Are You Ready? Intervention and Enrichment Worksheets 6 3 15. 2 32 Diagnose and Prescribe Multiply Decimals Are You Ready? CD-ROM Multiply. 18. 0.006 × 10 0.06 Are You Ready? Online Combine Like Terms Simplify each expression. 21. 6 + 3p + 14 + 9p 20 + 12p 22. 8y - 4x + 2y + 7x - x 10y + 2x 23. (12 + 3w - 5) + 6w - 3 - 5w 4 + 4w 24. 6n - 14 + 5n 11n - 14 Squares and Square Roots Tell whether each number is a perfect square. If so, identify its positive square root. 25. 42 no 26. 81 yes; 9 27. 36 yes; 6 28. 50 no 29. 100 yes; 10 30. 4 yes; 2 31. 1 yes; 1 32. 12 no Exponents and Polynomials NO INTERVENE A1NL11S_c07_0456-0459.indd 457 457 YES Diagnose and Prescribe ENRICH 6/25/09 9:05:06 AM ARE YOU READY? Intervention, Chapter 7 Prerequisite Skill Worksheets CD-ROM Exponents Skill 7 Activity 7 Evaluate Powers Skill 8 Activity 8 Multiply Decimals Skill 45 Activity 45 Combine Like Terms Skill 57 Activity 57 Squares and Square Roots Skill 6 Activity 6 Online Diagnose and Prescribe Online ARE YOU READY? Enrichment, Chapter 7 Worksheets CD-ROM Online Are You Ready? 457 CHAPTER Study Guide: Preview 7 Organizer Key Vocabulary/Vocabulario Objective: Help students Previously, you GI organize the new concepts they will learn in Chapter 7. <D @<I binomial binomio exponential expressions. degree of a monomial grado de un monomio simplified algebraic expressions by combining like terms. degree of a polynomial grado de un polinomio leading coefficient coeficiente principal monomial monomio perfect-square trinomial trinomio cuadrado perfecto polynomial polinomio scientific notation notación científica standard form of a polynomial forma estándar de un polinomio trinomial trinomio • wrote and evaluated • Online Edition Multilingual Glossary Resources You will study PuzzleView • properties of exponents. • powers of 10 and scientific Multilingual Glossary Online • KEYWORD: MA7 Glossary notation. how to add, subtract, and multiply polynomials by using properties of exponents and combining like terms. Answers to Vocabulary Connections Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. a method of writing very large and very small numbers 2. the coefficient of the first (lead) term, which is the number attached to the variable of that term You can use the skills in this chapter • to model area, perimeter, and volume in geometry. to express very small or very large quantities in science classes such as Chemistry, Physics, and Biology. 3. an expression with exactly 2 terms • 4. Possible answers: tricycle, triangle, tripod; they all have three of something; an expression with exactly three terms. • in the real world to model business profits and population growth or decline. 1. Very large and very small numbers are often encountered in the sciences. If notation means a method of writing something, what might scientific notation mean? 2. A polynomial written in standard form may have more than one algebraic term. What do you think the leading coefficient of a polynomial is? 3. A simple definition of monomial is “an expression with exactly one term.” If the prefix mono- means “one” and the prefix bi- means “two,” define the word binomial . 4. What words do you know that begin with the prefix tri-? What do they all have in common? Define the word trinomial based on the prefix tri- and the information given in Problem 3. 458 Chapter 7 A1NL11S_c07_0456-0459.indd 458 458 Chapter 7 6/25/09 9:05:26 AM CHAPTER 7 Organizer Reading Strategy: Read and Understand the Problem Objective: Help students apply Follow this strategy when solving word problems. strategies to understand and retain key concepts. • Read the problem through once. • Identify exactly what the problem asks you to do. GI • Read the problem again, slowly and carefully, to break it into parts. <D @<I Online Edition • Highlight or underline the key information. • Make a plan to solve the problem. ENGLISH LANGUAGE LEARNERS Reading Strategy: Read and Understand the Problem From Lesson 6-6 29. Multi-Step Linda works at a pharmacy for $15 an hour. She also baby-sits for $10 an hour. Linda needs to earn at least $90 per week, but she does not want to work more than 20 hours per week. Show and describe the number of hours Linda could work at each job to meet her goals. List two possible solutions. Step 1 Identify exactly what the problem asks you to do. Discuss Key information can include more than just numbers. Words and phrases such as not, no more than, whole number solutions, inches, and two different ways can be crucial to correctly solving a problem. • Show and describe the number of hours Linda can work at each job and earn at least $90 per week, without working more than 20 hours per week. Extend As you present word problems in Chapter 7 to the class, ask students what information should be highlighted and why. • List two possible solutions of the system. Step 2 Step 3 Break the problem into parts. Highlight or underline the key information. Make a plan to solve the problem. • Linda has two jobs. She makes $15 per hour at one job and $10 per hour at the other job. • She wants to earn at least $90 per week. • She does not want to work more than 20 hours per week. Answers to Try This 1a. Find the length and width of the rectangle. • Write a system of inequalities. • Solve the system. b. The difference between length and width is 14 units. The area is 120 square units. • Identify two possible solutions of the system. c. Write and solve a system of equations. Try This For the problem below, a. identify exactly what the problem asks you to do. b. break the problem into parts. Highlight or underline the key information. c. make a plan to solve the problem. 1. The difference between the length and the width of a rectangle is 14 units. The area is 120 square units. Write and solve a system of equations to determine the length and the width of the rectangle. (Hint: The formula for the area of a rectangle is A = w.) Exponents and Polynomials A1NL11S_c07_0456-0459.indd 459 Reading Connection 459 6/25/09 Powers of Ten by Philip and Phylis Morrison This is a wondrous yet practical guide to powers of 10 and exponential growth. The book begins with one billion light years. In 9:05:34 AM 1 40 steps, each step depicting _ 10 the scale of the previous step, it moves to subatomic particles. Activity Ask students to make their own books using 1 ft (10 0 ft) as step 6, 10 1 ft as step 7, 10 2 ft as step 8, and so on, up to step 11. Then 10 -1 ft will be step 5, 10 -2 ft will be step 4, etc. For each step, have students name (and perhaps locate a photo of) an object near that length. Reading and Writing Math 459 SECTION 7A Exponents One-Minute Section Planner Lesson Lab Resources Lesson 7-1 Integer Exponents • • □ Optional Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. NAEP ✔ SAT-10 ✔ ACT ✔ SAT ✔ SAT Subject Tests □ □ □ graphing calculator □ Lesson 7-2 Powers of 10 and Scientific Notation • • □ Optional Evaluate and multiply by powers of 10. Convert between standard notation and scientific notation. NAEP SAT SAT Subject Tests ✔ SAT-10 ✔ ACT □ □ □ Use patterns to explore multiplication properties of exponents. SAT-10 ✔ NAEP ACT SAT SAT Subject Tests □ □ □ graphing calculator □ 7-3 Algebra Lab Explore Properties of Exponents • □ Materials □ Algebra Lab Activities 7-3 Lab Recording Sheet Lesson 7-3 Multiplication Properties of Exponents • Use multiplication properties of exponents to evaluate and simplify expressions. ✔ SAT-10 □ ✔ NAEP □ ✔ ACT □ ✔ SAT □ ✔ SAT Subject Tests □ Lesson 7-4 Division Properties of Exponents • Use division properties of exponents to evaluate and simplify expressions. ✔ SAT-10 ✔ NAEP ✔ ACT ✔ SAT ✔ SAT Subject Tests □ □ □ □ Technology Lab Activities 7-4 Technology Lab □ Lesson 7-5 Rational Exponents • Evaluate and simplify expressions containing rational exponents. ✔ SAT-10 □ ✔ NAEP □ ✔ ACT ✔ SAT □ □ □ SAT Subject Tests Note: If NAEP is checked, the content is tested on either the Grade 8 or Grade 12 NAEP assessment. 460A Chapter 7 Optional graphing calculator, number cubes MK = Manipulatives Kit Math Background EXPONENTS Lesson 7-1 and to see that 10,000,000 is a power of 10, namely 10 7. Thus, 93,000,000 = 9.3 × 10 7. Up to this point, students have worked primarily with linear equations and linear inequalities. In Chapter 7, students move toward more complex ideas as they begin to study polynomials. Before beginning this study, students must first have an understanding of exponents. In general, every positive real number may be written in scientific notation, a × 10 n, where 1 ≤ a < 10 and n is an integer. The value of a is called the coefficient. A number line helps to visualize scientific notation. On the number line below, several powers of 10 are graphed. One common difficulty students have with exponents is the use of zero. For example, students are often puzzled by the fact that any nonzero number raised to the zero power is 1. It makes sense to think of 2 4 as a product where 2 is a factor 4 times (2 4 = 2 · 2 · 2 · 2), but when it comes to evaluating 2 0, how does one write a product with 2 as a factor zero times? Students should understand that 2 0 is defined to be 1 in order to make it consistent with the rules of exponent arithmetic. For example, in order for the Quotient of Powers Property to work in as many situations as 24 = 2 4-4 = 2 0, possible, it must be true that __ 24 4 2 but __4 = 1. Thus, 2 0 = 1. The interval between each successive power of 10 is 10 times as large as the preceding interval. When a number is written in scientific notation, such as 6.7 × 10 3, the power of 10 tells which interval the number lies in and the coefficient tells where the number falls within the interval. 101 102 103 10 1,000 100 6.7 × 103 104 2 This idea of defining certain powers in order to create a system that is as widely consistent as possible also explains why the expression 0 0 is undefined. First, it is clear that 0 1 = 0, 0 2 = 0, and 0 13 = 0. In fact, for any value of n greater than zero, 0 n = 0. For this reason, it might make sense to define 0 0 as zero. On the other hand, as shown above, any nonzero number raised to the zero power is 1, so it might also make sense to define 0 0 as 1. Because there is no single real number that works consistently as a definition of 0 0, this expression is considered indeterminate and is left undefined. SCIENTIFIC NOTATION LESSON 7-2 10,000 Numbers greater than or equal to 1 but less than 10 are written in scientific notation with an exponent of 0 since, for example, 3.8 = 3.8 × 1 = 3.8 × 10 0 Numbers greater than 0 but less than 1 are written with negative powers of 10. A specific example shows why this is the case: 0.0041 = 4.1 × 0.001 1 = 4.1 × 1000 = 4.1 × 10 -3 _ Scientific notation is an efficient way to write very large and very small numbers, such as numbers used to express distances in space. For example, the distance from the Earth to the Sun is approximately 93 million miles or 93,000,000 miles. The key step in the translation to scientific notation is to recognize that 93,000,000 = 9.3 × 10,000,000 460B 7-1 Organizer 7-1 Pacing: Traditional 1 day Block Integer Exponents __1 day 2 Objectives: Evaluate expressions containing zero and integer exponents. GI Simplify expressions containing zero and integer exponents. <D @<I Objectives Evaluate expressions containing zero and integer exponents. Who uses this? Manufacturers can use negative exponents to express very small measurements. Simplify expressions containing zero and integer exponents. In 1930, the Model A Ford was one of the first cars to boast precise craftsmanship in mass production. The car’s pistons had a diameter of 3 _78_ inches; this measurement could vary by at most 10 -3 inch. Online Edition Tutorial Videos Countdown Week 15 You have seen positive exponents. Recall that to simplify 3 2, use 3 as a factor 2 times: 3 2 = 3 · 3 = 9. But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out. Warm Up Base x4 Evaluate each expression for the given values of the variables. Exponent Power 55 54 53 52 51 Value 3125 625 125 25 5 ÷5 1. x3 y2 for x = -1 and y = 10 -100 3x2 2. _ for x = 4 and y = -7 y2 48 49 Write each number as a power of the given base. 5 =1 50 = _ 5 5 -2 ÷5 1 1 =_ 1 ÷5=_ 5 -2 = _ 5 25 5 2 1 1 =_ 5 -1 = _ 5 51 Integer Exponents 43 4. -27; base -3 ÷5 5 -1 When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5: _ 3. 64; base 4 ÷5 50 WORDS (-3)3 NUMBERS Zero exponent—Any nonzero number raised to the zero power is 1. Also available on transparency Negative exponent—A nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent. 2 -4 is read “2 to the negative fourth power.” The Babylonian number system had no symbol for zero. It was represented by a blank space. 3 = 1 123 = 1 0 (-16) 0 = 1 0 ALGEBRA If x ≠ 0, then x 0 = 1. (_37 ) = 1 0 1 =_ 1 3 -2 = _ 9 32 1 =_ 1 2 -4 = _ 16 24 If x ≠ 0 and n is an integer, 1. then x -n = _ xn Notice the phrase “nonzero number” in the table above. This is because 0 0 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0 0 = __00 . 1 Also, 0 -6 would be __ = _10_. Since division by 0 is undefined, neither value exists. 6 0 460 Chapter 7 Exponents and Polynomials 1 Introduce A1NL11S_c07_0460-0465.indd e x p l o r460 at i o n Motivate 7-1 Integer Exponents A botanist has taken over a study of a plant whose height doubles every day. On the fourth day after the experiment began, the plant was 16 inches tall. 1. The botanist can find the height of the plant on previous days by repeatedly dividing by 2. Use this fact to complete the middle column of the table. 2. Use your knowledge of exponents to fill in the right column for days 1 and 2. Then look for patterns to complete the right column. 3. What does day 0 represent? What was the height of the plant on day 0? How can you write the height as a power? KEYWORD: MA7 Resources Day Height Height of Plant Written as (in.) a Power 4 4 16 2 3 8 23 2 1 0 ⫺1 ⫺2 ⫺3 4. What does day ⫺1 represent? What was the height of the plant on day ⫺1? How can you write the height as a power? THINK AND DISCUSS 460 Chapter 7 5. Describe the pattern in the right column of the table. 6. Show how you could find the height of the plant on day ⫺4 and then write the height as a power. Show students the following examples and ask them to suggest a rule about the use of negative exponents. 1 1 1 a. 2-1 = _ b. 2-3 = _ c. 5-2 = _ 2 25 8 1 d. 3-2 = _ 9 Possible answer: The negative exponent means that you must use the reciprocal of the base and change the exponent to a positive number. Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 6/25/09 9:06:08 AM EXAMPLE 1 Manufacturing Application "" The diameter for the Model A Ford piston could vary by at most 10 -3 inch. Simplify this expression. Students will sometimes multiply the base by the negative exponent. Have these students reread the definition of a negative exponent. Point out that 10-3 means a number less than one, not a number less than zero. 1 =_ 1 1 10 -3 = _ =_ 10 3 10 · 10 · 10 1000 1 inch, or 0.001 inch. 10 -3 inch is equal to ____ 1000 A sand fly may have a wingspan up to 5 -3 m. Simplify this 1 expression. 1. Ê,,", ,/ _m 125 EXAMPLE 2 Zero and Negative Exponents Additional Examples Simplify. A 2 -3 1 =_ 1 1 2 -3 = _ =_ 23 2 · 2 · 2 8 5 0 = 1 Any nonzero number raised to the zero power is 1. C (-3)-4 In (-3) -4, the base is negative because the negative sign is inside the parentheses. In -3 -4 the base (3) is positive. Simplify. 1 B. 7 0 1 A. 4-3 _ 64 1 1 C. (-5)-4 _ D. -5-4 - _ 625 625 D -3 -4 1 = -_ 1 1 -3 -4 = - _ = -_ 3·3·3·3 81 34 Example 3 1 1 _ 2b. (-2) _ -4 10,000 3 16 _ _ 32 32 2c. (-2)-5- 1 2d. -2 -5 - 1 Evaluating Expressions with Zero and Negative Exponents Evaluate each expression for the given value(s) of the variable(s). A x -1 for x = 2 2 -1 2 -1 Substitute 2 for x. 1 =_ 1 =_ 21 2 1 1·_ 1 __ Questioning Strategies Write the power in the denominator as a product. (-2)(-2)(-2) _ _ INTERVENTION Simplify expressions with exponents. (-2)3 3a. 1 64 3b. 2 1· 1 -8 1 -_ 8 3a. p • What is the difference between Examples 2C and 2D? Simplify. • When will a term with a negative exponent have a negative value? -2 0 for p = 4 3b. 8a b for a = -2 and b = 6 7-1 Integer Exponents A1NL11S_c07_0460-0465.indd 461 3 • There are no fractions in the problem. Why are there fractions in the answers? Inclusion In Example 3, some students may prefer to rewrite the expression with a positive exponent before substituting for the variable. 6/25/09 9:06:15 AM Guided Instruction Technology Students can use the or Yx keys on their calculators to check their work in Examples 1—3. 461 EX AM P LE • Why is the value of a variable irrelevant if that variable is raised to the zero power? 2 Teach Explain negative and zero exponents by demonstrating the pattern in the decreasing values of the powers. Use the definition to simplify and evaluate expressions with negative and zero exponents. Then include examples with negative exponents in the denominator. Remind students that factors with negative exponents should be simplified. EX AM P LES 1 – 2 Simplify the power in the denominator. Evaluate each expression for the given value(s) of the variable(s). -3 Evaluate each expression for the given value(s) of the variable(s). 1 A. x -2 for x = 4 _ 16 B. -2a 0b-4 for a = 5 and 2 b = -3 - _ 81 1 . Use the definition x -n = _ xn B a 0b -3 for a = 8 and b = -2 8 0 · (-2)-3 Substitute 8 for a and -2 for b. 1· One cup is 2-4 gallons. Simplify 1 this expression. _ gal 16 Example 2 1 = __ 1 1 (-3)-4 = _ =_ (-3) 4 (-3)(-3)(-3)(-3) 81 Simplify. 2a. 10 -4 EXAMPLE Example 1 B 50 Through Graphic Organizers Have students make a chart similar to the one below and let them refer to it during class work and homework. 0 x (x ≤ 0) Undefined x 0 (x ≠ 0) 1 x -2 (x ≠ 0) 1 _ x2 1 _ (x ≠ 0) x -2 x2 Lesson 7-1 461 What if you have an expression with a negative exponent in a denominator, 1 ? such as ___ -8 x Additional Examples x Example 4 Simplify. 7 A. 7w -4 _4 w a0b -2 C. _ c -3d 6 -5 B. _ k-2 -n 1 = x -n 1 , or _ =_ xn xn 1 = x -(-8) _ x -8 Definition of negative exponent Substitute -8 for n. = x8 -5k2 Simplify the exponent on the right side. So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator. c3 _ 2 b d6 An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. EXAMPLE INTERVENTION 4 Simplifying Expressions with Zero and Negative Exponents Simplify. Questioning Strategies -4 B _ -4 A 3y -2 EX A M P L E 4 k -4 = -4 · 1 _ k -4 k -4 _ 3y -2 = 3 · y -2 • How do you decide which factors get moved to the other side of the fraction bar? _ = 3 · 12 y 3 =_ y2 • What happens to factors with exponents of zero? = -4 · k 4 = -4k 4 x -3 C _ 0 5 a y x -3 = _ 1 _ a 0y 5 x 3 · 1 · y 5 1 =_ x 3y 5 Simplify. 4a. 2r 0m -3 1. a 0 = 1 and x -3 = _ x3 2 _ m 3 1 r -3 _ 4b. _ 7 7r 3 g4 4c. _ g 4h 6 h -6 THINK AND DISCUSS -3 s =_ 2,_ 1 , ? -2 = _ 1 1. Complete each equation: 2b ? = _ b2 k ? s3 t2 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe how to simplify, and give an example. -«vÞ}ÊÝ«ÀiÃÃÃÊÜÌ Ê i}>ÌÛiÊÝ«iÌÃ ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ ÊÌ iÊÕiÀ>ÌÀÊ°Ê°Ê° 462 Chapter 7 Exponents and Polynomials Answers to Think and Discuss 3 Close Summarize 1. -2; 0; t A1NL11S_c07_0460-0465.indd 462 and INTERVENTION Remind students that an expression is not simplified if it has an exponent that is negative or zero. Diagnose Before the Lesson 7-1 Warm Up, TE p. 460 Have students state the rules for simplifying expressions with negative exponents in their own words. Accept nontechnical answers such as the following: Make the exponent positive and move the factor to the other side of the fraction bar. Monitor During the Lesson Check It Out! Exercises, SE pp. 461–462 Questioning Strategies, TE pp. 461–462 462 Chapter 7 ÀÊ>Êi}>ÌÛiÊiÝ«iÌÊ ÊÌ iÊ`i>ÌÀÊ°Ê°Ê° Assess After the Lesson 7-1 Lesson Quiz, TE p. 465 Alternative Assessment, TE p. 465 2. See p. A6. 7/18/09 4:51:41 PM 7-1 Exercises 7-1 Exercises KEYWORD: MA7 7-1 KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 p. 461 Assignment Guide 1. Medicine A typical virus is about 10 -7 m in size. Simplify this expression. 1 _ m 10,000,000 SEE EXAMPLE 2 Simplify. 1 _ 3. 3 1 36 1 1 8. 10 _ 7. -8 - _ 2. 6 -2 p. 461 -2 100 512 SEE EXAMPLE 3 p. 461 _ _ _ 12. b -2 for b = -3 1 13. (2t)-4 for t = 2 _ 1 14. (m - 4)-5 for m = 6 32 p. 462 If you finished Examples 1–2 Basic 24—36, 77, 86, 88 Average 24—36, 77, 86, 88 Advanced 24—36, 77, 100, 101 Evaluate each expression for the given value(s) of the variable(s). 9 SEE EXAMPLE 4 _ _ 1 - 1 -3 6. 1 -8 1 25 5. 3 27 0 -3 9. (4.2) 1 10. (-3) - 1 11. 4-2 1 16 27 4. -5 -2 0 -3 Assign Guided Practice exercises as necessary. 20. 2x 0y -4 17. 3k -4 y _3 k4 _ _2 6 f -4 g 21. _ g -6 f 4 4 256 15. 2x 0y -3 for x = 7 and y = -4 - Simplify. 16. 4m 0 4 1 _ If you finished Examples 1–4 Basic 24—77, 86—99, 102—113 Average 24—57, 58—74 even, 76—100, 102—113 Advanced 24—57, 58—74 even, 77—113 1 _ 32 7 18. _ 7r 7 r -7 x 10 19. _ x 10d 3 d -3 c4 22. _ c 4d 3 d -3 23. p 7q -1 p _ 7 q Homework Quick Check Quickly check key concepts. Exercises: 24, 28, 34, 42, 52, 77 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 24 25–36 37–42 43–57 1 2 3 4 Extra Practice Skills Practice p. S16 Application Practice p. S34 24. Biology One of the smallest bats is the northern blossom bat, which is found from Southeast Asia to Australia. This bat weighs about 2 -1 ounce. Simplify this expression. 1 oz _ 2 Simplify. 26. 5 -4 25. 8 0 1 1 _ 625 1 _ _ 49 36 1 34. (-4) 16 33. (-3) - _ 29. -6 -2 - 1 30. 7 -2 -1 2 3 1 _ 27. 3 -4 81 2 01 31. _ (5) 1 35. (_ 2) -2 4 1 _ 81 1 _ 169 1 -_ 28. -9 -2 32. 13 -2 36. -7 -1 7 Evaluate each expression for the given value(s) of the variable(s). 37. x -4 for x = 4 1 _ 38. 256 39. (10 - d) for d = 11 1 0 43. k -4 _1 k4 44. 2z -8 x 6y 2 53. b 0c 0 1 for v = 9 40. 10m n 1 _ _2 z8 0 _ 7 1 _ 216 for m = 10 and n = -2 - 32 _ _s r5 _ 46. c -2d _d c 2 s 5 s 5t 12 51. _ t -12 47. -5x -3 - _5 x3 6 3w -5 3x 52. _ -6 5 x w _ _ 3 -1 q -2r 0 1 h3 a -7b 2 k 2 m -1n 5 55. _ _ 54. _ 56. _ 57. h 5 2 4 0 3 -4 2 2n 3 s c d b d 6m 6m 2k q2 _ 3m _ a 7c 3 7-1 Integer Exponents A1NL11S_c07_0460-0465.indd 463 1 _ 42. 4w vx v for w = 3, v = 0, and x = -5 4 3 1 b 45. _ 2b -3 2 2f r -5 _ 48. 4x -6y -2 4 49. _ 2g 10 50. s -1 7g -10 _ -3 -1 -5 -2 1 and b = 8 41. (3ab) for a = _ 2 144 Simplify. (_23 v) 463 6/25/09 9:06:34 AM KEYWORD: MA7 Resources Lesson 7-1 463 Evaluate each expression for x = 3, y = -1, and z = 2. Exercise 94 involves writing an equation with a negative exponent. This exercise prepares students for the Multi-Step Test Prep on page 494. _1 58. z -5 62. 66. 59. 1 _ 1 77. red blood cell: ______ m; white 125,000 3 m; platelet: blood cell: ______ 250,000 Simplify. 61. (xyz)-1 - _1 1 6 .q a _ 68. c 3 67. a 3b -2 Biology , * XXXXXXX , . .q , XXXXX q, .q d _ 3 -4 d3 _ _ _ 4 _ _ 77. Biology Human blood contains red blood cells, white blood cells, and platelets. The table shows the sizes of these components. Simplify each expression. 23 = 8, 22 = 4, 21 = 2, 20 = 1, 1 1 2-1 = _, 2-2 = _, 4 2 1 1 2-3 = _ = _ 8 23 60. (yz) 0 1 16 0 5 2 69. v 0w 2y -1 w 70. (a 2b -7) 1 71. -5y -6 c b2 y6 3 y -5 3 2 -8 20p -1 4q 6 2a 2a m x 3 _ _ _ _ _ 1 76. 2b 73. -1 2a b 74. -3 72. 75. p b -6 b n 3y 12 5q -3 m 2n 3 8 12 5 3x y a 3 ________ m 1,000,000 85. Possible answer: Look at the pattern below. As the exponent goes down by 1, the value is half of what it was before. 1 _ Explain the error. 66. Equation A is incorrect because 5 was incorrectly moved to the denom. The neg. exp. applies only to the base x. Answers (x + y)-4 63. x -y 3 64. (yz) -x - _ 65. xy -4 3 (xy - 3)-2 36 8 /////ERROR ANALYSIS///// Look at the two equations below. Which is incorrect? 32 Blood Components Part 125,000 -1 Red blood cell Tell whether each statement is sometimes, always, or never true. When bleeding occurs, platelets (which appear green in the image above) help to form a clot to reduce blood loss. Calcium and vitamin K are also necessary for clot formation. Size (m) White blood cell 3(500)-2 Platelet 3(1000)-2 1 78. If n is a positive integer, then x -n = __ . always xn 79. If x is positive, then x -n is negative. never 80. If n is zero, then x -n is 1. sometimes 81. If n is a negative integer, then x -n = 1. sometimes 82. If x is zero, then x -n is 1. never 83. If n is an integer, then x -n > 1. sometimes 84. Critical Thinking Find the value of 2 3 · 2 -3. Then find the value of 3 2 · 3 -2. Make a conjecture about the value of a n · a -n. 1; 1; a n · a -n = 1 1 85. Write About It Explain in your own words why 2 -3 is the same as __ . 3 2 Find the missing value. 1 =2 86. _ 4 -2 1 90. 7 -2 = _ 49 1 87. 9 -2 = _ 81 91. 10 1 = 88. _ 64 -2 89. _ = 3 -1 1 3 1 =2·5 93. 2 · _ -1 5 8 3 1 =_ -3 92. 3 · 4 -2 = _ 16 1000 94. This problem will prepare you for the Multi-Step Test Prep on page 494. a. The product of the frequency f and the wavelength w of light in air is a constant v. Write an equation for this relationship. f w = v b. w = v ; w = v f -1 f b. Solve this equation for wavelength. Then write this equation as an equation with f raised to a negative exponent. c. The units for frequency are hertz (Hz). One hertz is one cycle per second, which -1 is often written as __1s . Rewrite this expression using a negative exponent. s _ 7-1 PRACTICE A 7-1 PRACTICE C _______________________________________ ___________________ __________________ Practice B 7-1 PRACTICE B Integer Exponents LESSON 7-1 464 Chapter 7 Exponents and Polynomials Simplify. 1. 53 = 3. (5)2 5. 60 1 53 1 25 = 1 2. 26 = 125 4. (4)3 1 6. (7)2 1 26 Name _______________________________________ Date___________________ Class __________________ 1 = 64 8. a5b6 for a = 3 and b = 2 1 8 1 64 1 49 10. 5zx for z = 3 and x = 2 11. (5z)x for z = 3 and x = 2 5 9 1 9 A1NL11S_c07_0460-0465.indd 12. c3 (162) for c = 4 1 225 1 16,384 Simplify. 13. t4 14. 3r5 s 3 t 5 5 1 3 t t4 r5 s3 2x 3 y 2 17. z4 h0 16. 3 1 3 19. 15. 4fh 20. 7c a 4 c 2e 0 b 1d 3 a4 bc2 d 3 10a 4 b 21. k 5 = 1 k5 2. What number can go in the box to make a true statement: 5 1 3. Write the expression 3 with a negative exponent. 8 3g 2 hk 2 6h0 Chapter 7 1 =1 2 53 = 0 31 24 = 5. 25 32 100,000 7. 70 1 inch or 0.125 inch 8 9. ( 4)3 8. 13. 8x 64 1 5 10. 1 106 1,000,000 1 ( 4)3 64 c2 2 3 12. c d t4 8 x5 8 b 1 32 6. 25 1 -3 1 Simplify each expression. h 2g 2 k 2 14. 12r d3 0 12 1 = 82 82 1 24 1 = 24 24 Simplify 4 2. Simplify x2y 3z0. x2y3z0 Write without negative exponents. x 2 z0 y3 Write in expanded form. x 2 (1) y3 z0 = 1. Simplify. x2 y3 Simplify. Write without negative exponents. 6 0 = 1? 4. What is the reciprocal of b7? Negative Exponents in the Denominator 42 1 42 1. What is the base of the expression 6 ? 3 11. t 4 464 0 60 = 1 Negative Exponents For any nonzero number x For any nonzero number 1 x and any integer n, and any integer n, xn = n . 1 x = xn. x n For any nonzero number x, x0 = 1. 00 and 0n are undefined. 4 23. A ball bearing has diameter 23 inches. Evaluate this expression. Examples 1 = m3 m 3 22.A cooking website claims to contain 105 recipes. Evaluate this expression. Zero Exponents Definition Answer each question. 5g 5 x 3 y 2z 4 14a 4 20bc 1 7-1 RETEACH Integer Exponents Positive exponents: The answer is the base multiplied by itself the number of times identified by the exponent. Zero exponent: The answer is always 1 (if the base is not 0; 00 is undefined). 1 1 3 –2 = = 3•3 9 Negative exponents: The answer is 1 1 3 –3 = = the reciprocal of the same expression 3 • 3 • 3 27 with a positive exponent. 1 1 3 –4 = = 3 • 3 • 3 • 3 81 Note that the rules are the same when the base is a variable: g0 = 1 Review for Mastery 7-1 Remember that 23 means 2 2 2 = 8. The base is 2, the exponent is positive 3. Exponents can also be 0 or negative. 30 = 1 1 3 –1 = 3 b3 = b • b • b 4fg 5 18. 5h 3 2 34 = 3 • 3 • 3 • 3 = 81 33 = 3 • 3 • 3 = 27 32 = 3 • 3 = 9 46431 = 3 Name _______________________________________ Date___________________ Class __________________ LESSON Studying the patterns that are found in expressions with exponents can help you remember the rules for evaluating expressions with integer exponents. 9. (b 4)2 for b = 1 243 64 7-1 READING STRATEGIES Using Patterns 7-1 Evaluate each expression for the given value(s) of the variable(s). 7. d3 for d = 2 Reading Strategies LESSON 7 1 4•4 1 16 Fill in the blanks to simplify each expression. 1. 25 2. 103 25 = 1 3. 103 = 5 2 1 1 = 25 2•2•2•2•2 1 1 = 103 10 • 10 • 10 1 = 32 1 54 1 4 = 5 54 1 10 5 4 1 = 1000 = 5•5•5•5 = 625 Simplify. 4. 5y 4 5 y4 5. x3 7. 1 x y x4 y b2 8. 1 3 a b 8 a 3 8a 3 a b 6. 9x 3 y 2 4 9. 5x y 2 9x 3 y2 5y 2 x4 6/25/09 9:06:41 AM In Exercise 96, if students chose F, they may have used the negative exponent as a factor. If they chose G, they may have made the base negative because the exponent is negative. 95. Which is NOT equivalent to the other three? 1 _ 25 5 -2 0.04 -25 (-6)(-6) 1 -_ 6·6 1 _ 6·6 a 3b 2 _ -c a3 _ -b 2c c _ a 3b 2 96. Which is equal to 6 -2? 6 (-2) If students chose D in Exercise 97, they probably moved every factor to the opposite side of the fraction bar, even if it had a positive exponent. 3 -2 ab . 97. Simplify _ c -1 a 3c _ 2 b _5 , or 1.25 98. Gridded Response Simplify ⎡⎣2 -2 + (6 + 2)0⎤⎦. 4 99. Short Response If a and b are real numbers and n is a positive integer, write a simplified expression for the product a -n · b 0 that contains only positive exponents. 1 ; a -n = 1 and b 0 = 1 for b ≠ 0. Explain your answer. _ CHALLENGE AND EXTEND 100. rapidly as x increases. -4 1 16 x __ y = 2x x _ 100. Multi-Step Copy and complete the table of values below. Then graph the ordered pairs and describe the shape of the graph. Possible answer: y increases more y _ _ an an So you have 1n · 1, or simply 1n . a a -3 1 8 -2 1 4 __ -1 1 2 __ __ Journal 0 1 2 3 4 1 2 4 8 16 101. Multi-Step Copy and complete the table. Then write a rule for the values of 1 n and n n (-1)n when n is any negative integer. 1 n = 1; (-1) = -1 if n is odd, and (-1) = Have students use patterns to explain why any number raised to the zero power, except zero, is one. 1 if n is even. -1 -2 1n 1 1 1 1 1 (-1)n -1 1 -1 1 -1 n -3 -4 -5 Have students choose three exercises from Exercises 37—42, write each expression in words, and then show two different ways to evaluate each expression. SPIRAL REVIEW Solve each equation. (Lesson 2-3) y 104. _ - 8 = -12 -20 5 102. 6x - 4 = 8 2 103. -9 = 3 (p - 1) -2 105. 1.5h - 5 = 1 4 1 n + 2 - n 28 106. 2w + 6 - 3w = -10 16 107. -12 = _ 2 Identify the independent and dependent variables. Write a rule in function notation for each situation. (Lesson 4-3) 110. y = 3x - 4 111. y = 1 x + 5 3 112. y = 2 3 113. y = -4x + 9 _ _ 108. Pink roses cost $1.50 per stem. ind.: number of roses; dep.: total cost; f (x) = 1.50x 109. For dog-sitting, Beth charges a $30 flat fee plus $10 a day. ind.: number of days; dep.: total cost; f (x) = 10x + 30 Write the equation that describes each line in slope-intercept form. (Lesson 5-7) 110. slope = 3, y-intercept = -4 111. slope = _13_, y-intercept = 5 112. slope = 0, y-intercept = __23 113. slope = -4, the point (1, 5) is on the line. 7-1 Integer Exponents ________________________________________ LESSON 7-1 ___________________ __________________ Problem Solving 7-1 PROBLEM SOLVING Integer Exponents Write the correct answer. 1. At the 2005 World Exposition in Aichi, Japan, tiny mu-chips were embedded in the admissions tickets to prevent counterfeiting. The mu-chip was developed by Hitachi in 2003. Its area 42(10)−2 square millimeters. Simplify A1NL11S_c07_0460-0465.indd is465 this expression. 2. Despite their name, Northern Yellow Bats are commonly found in warm, humid areas in the southeast United States. An adult has a wingspan of about 14 inches and weighs between 3(2)−3 and 3(2)−2 ounces. Simplify these expressions. 3 3 and oz 4 8 4 or 0.16 mm 2 25 3. Saira is using the formula for the area of a circle to determine the value of π. She is using the expression Ar−2 where A = 50.265 and r = 4. Use a calculator to evaluate Saira’s expression to find her approximation of the value of π to the nearest thousandth. 4. The volume of a freshwater tank can be expressed in terms of x, y, and z. Expressed in these terms, the volume of the tank is x3y−2z liters. Determine the volume of the tank if x = 4, y = 3, and z = 6. 42 3.142 2 liters 3 Alison has an interest in entomology, the study of insects. Her collection of insects from around the world includes the four specimens shown in the table below. Select the best answer. Insect Mass Emperor Scorpion 2−5 kg African Goliath Beetle 11−1 kg Giant Weta 2−4 kg Madagascar Hissing Cockroach 5−3 kg A − 6. Many Giant Wetas are so heavy that they cannot jump. Which expression is another way to show the mass of the specimen in Alison’s collection? F −(2)4 kg § 1· G ¨ ¸ ©2¹ H 1 kg 2•2•2•2 −4 kg J 4 1 kg 2 1 kg 125 1 kg 125 C 1 kg 15 D 125 kg 7. Scorpions are closely related to spiders and horseshoe crabs. What is the mass of Alison’s Emperor Scorpion expressed as a quotient? 1 kg A − 32 B 1 kg 25 LESSON 7-1 C 1 kg 32 Challenge ___________________ __________________ 7-1 CHALLENGE Exploring Patterns in the Units Digit of xn When you write out the first several powers of xn, where x and n are positive integers, you can discover interesting patterns in the units digits of xn. x1 x=2 21 = 2 1 x2 x3 x4 22 = 2(2) = 4 23 = 2(4) = 8 24 = 2(8) = 16 x5 5 2 x6 25 = 2(16) = 32 26 = 2(32) = 64 6 Notice that 2 and 2 have the same units digit and that 2 and 2 have the same units digit. In the exercises that follow, you can discover other number patterns involving the units digits of xn. 1. x=1 2. x=2 3. x=3 4. x=4 5. x=5 6. x=6 7. x=7 x=8 9. x=9 10. x = 10 x1 x2 x3 x4 x5 x6 x7 x8 x9 1 2 3 4 5 6 7 8 9 0 1 4 9 6 5 6 9 4 1 0 1 8 7 4 5 6 3 2 9 0 1 6 1 6 5 6 1 6 1 0 1 2 3 4 5 6 7 8 9 0 1 4 9 6 5 1 8 7 4 5 6 3 2 9 0 1 6 1 6 5 6 1 6 1 0 1 2 3 4 5 6 7 8 9 0 9 4 1 0 465 1. A square foot is 3-2 square yards. Simplify this 1 expression. _ yd2 9 Simplify. 1 2. 2 -6 _ 64 1 3. (-7)-3 - _ 343 5. -112 -121 4. 60 1 Evaluate each expression for the given value(s) of the variable(s). 1 6. x-4 for x = 10 _ 10,000 7. 2a -1b -3 for a = 6 and 1 b=3 _ 81 Simplify. -3 4 9. _ -3y 6 8. 4y -5 _5 y y -6 7/18/09 4:52:07 PM In Exercises 1–10, find the first nine powers of each value of x. Using the units digit of each result, complete the table. You may find a calculator useful. 8. 5. Cockroaches have been found on every continent, including Antarctica. What is the mass of Alison’s Madagascar Hissing Cockroach expressed as a quotient? B ________________________________________ 7-1 x -4 10. _ a 0y 3 1 _ x 4y 3 Also available on transparency Refer to the table that you completed in Exercises 1–10. Describe the pattern in the units digits of xn. 11. 1n For all n, 1n has 1 as its units digit. 12. 2n The pattern is 2, 4, 8, and 6, for n = 1, 2, 3, and 4 and then repeats. 13. 3n The pattern is 3, 9, 7, and 1, for n = 1, 2, 3, and 4 and then repeats. 14. 5n For all n > 0, 5 n has 5 as its units digit. 15. Write a rule that determines the units digit of 7n as a function of n. If you divide n by 4, then the units digit is 7, 9, 3, or 1, depending on whether the remainder is 1, 2, 3, or 0, respectively. D 32 kg Lesson 7-1 465 7-2 Organizer 7-2 Pacing: Traditional 1 day Block __1 day Powers of 10 and Scientific Notation 2 Objectives: Evaluate and multiply by powers of 10. @<I Online Edition Tutorial Videos, Interactivity The table shows relationships between several powers of 10. Vocabulary scientific notation Countdown Week 15 ÷ 10 10 3 10 Value 1000 100 2. 123 ÷ 1000 0.123 3. 0.003 × 100 0.3 4. 0.003 ÷ 100 0.00003 5. 104 7. 230 ÷ 10 ÷ 10 10 1 10 0 10 -1 10 -2 10 -3 10 1 1 = 0.1 _ 10 1 = 0.01 _ 100 1 = 0.001 _ 1000 WORDS NUMBERS Positive Integer Exponent 0.0001 10 4 = 1 0, 0 0 0 If n is a positive integer, find the value of 10 n by starting with 1 and moving the decimal point n places to the right. 1 4 places Negative Integer Exponent Also available on transparency If n is a positive integer, find the value of 10 -n by starting with 1 and moving the decimal point n places to the left. EXAMPLE 1 Q: How did the number written in scientific notation feel after he changed into standard form? A: Powerless. ÷ 10 Powers of 10 10,000 6. 10-4 ÷ 10 × 10 × 10 × 10 × 10 × 10 × 10 • Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left. • Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right. Evaluate each expression. 123,000 2 Power Warm Up 1. 123 × 1000 ÷ 10 Nucleus of a silicon atom ⎧ ⎨ ⎩ <D Convert between standard notation and scientific notation. 1 = 0.0 0 0 0 0 1 10 -6 = _ 10 6 ⎧ ⎨ ⎩ GI Convert between standard notation and scientific notation. Why learn this? Powers of 10 can be used to read and write very large and very small numbers, such as the masses of atomic particles. (See Exercise 44.) Objectives Evaluate and multiply by powers of 10. 6 places Evaluating Powers of 10 Find the value of each power of 10. A 10 -3 You may need to add zeros to the right or left of a number in order to move the decimal point in that direction. 466 B 10 2 C 10 0 Start with 1 and move the decimal point three places to the left. Start with 1 and move the decimal point two places to the right. Start with 1 and move the decimal point zero places. 0. 0 0 1 1 0 0 1 0.001 100 Chapter 7 Exponents and Polynomials 1 Introduce A1NL11S_c07_0466-0471.indd e x p l o r466 at i o n 7-2 Powers of 10 and Scientific Notation You will need a calculator for this Exploration. 1. You can use the exponent key, , on your calculator to evaluate powers of 10. Use your calculator as needed to complete the table. Power of 10 Value 10 5 10 6 10 7 10 8 10 9 2. Look for patterns in the table. How is the exponent in each power of 10 related to the value of that power of 10? 3. What happens when you try to use your calculator to evaluate larger powers of 10, such as 1015? Motivate Have students copy the following numbers: 0.0000000000095 2,700,000,000,000,000,000,000 Ask them why the numbers are difficult to copy accurately. They have many zeros. Say that numbers used in science and technology often contain many zeros. Scientific notation was developed to make these numbers easier to work with. THINK AND DISCUSS KEYWORD: MA7 Resources 4. Explain how you could write the value of 1015. How many zeros would you write? 5. Describe a general rule you can use to write the value of 10 n, where n is a positive integer. 466 Chapter 7 Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 6/25/09 9:08:50 AM Find the value of each power of 10. 1a. 10 -2 0.01 1b. 10 5 100,000 "" 1c. 10 10 10,000,000,000 EXAMPLE 2 When writing the powers of 10 in a chart or a list, students will often start at 10, rather than 1. Remind students that 100 = 1. Writing Powers of 10 Write each number as a power of 10. A 10,000,000 If you do not see a decimal point in a number, it is understood to be at the end of the number. B 0.001 C 10 The decimal point is seven places to the right of 1, so the exponent is 7. The decimal point is three places to the left of 1, so the exponent is -3. The decimal point is one place to the right of 1, so the exponent is 1. 10 7 10 -3 10 1 Write each number as a power of 10. 2a. 100,000,000 10 8 2b. 0.0001 10 -4 Ê,,", ,/ Additional Examples Example 1 Find the value of each power of 10. 2c. 0.1 10 -1 A. 10-6 0.000001 You can also move the decimal point to find the product of any number and a power of 10. You start with the number instead of starting with 1. Multiplying by Powers of 10 125 × 10 5 = 12,5 0 0, 0 0 0 If the exponent is a negative integer, move the decimal point to the left. 36.2 × 10 -3 = 0.0 3 6 2 10,000 C. 109 1,000,000,000 Example 2 ⎧ ⎨ ⎩ If the exponent is a positive integer, move the decimal point to the right. B. 104 Write each number as a power of 10. 5 places ⎧ ⎨ ⎩ A. 1,000,000 3 places 10-4 B. 0.0001 EXAMPLE 3 103 Multiplying by Powers of 10 C. 1000 Find the value of each expression. Example 3 A 97.86 × 10 6 97.8 6 0 0 0 0 Find the value of each expression. Move the decimal point 6 places to the right. 97,860,000 B 19.5 × 10 -4 0 0 1 9.5 106 Move the decimal point 4 places to the left. A. 23.89 × 108 2,389,000,000 B. 467 × 10-3 0.467 0.00195 Find the value of each expression. 3a. 853.4 × 10 5 85,340,000 3b. 0.163 × 10 -2 0.00163 INTERVENTION Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied. The first part is a number that is greater than or equal to 1 and less than 10. Questioning Strategies EX AM P LE 1 • What does a positive exponent represent? • What does a negative exponent represent? The second part is a power of 10. EX AM P LE 7- 2 Powers of 10 and Scientific Notation 2 Teach 6/25/09 9:08:59 AM Guided Instruction Show students the pattern of powers of 10. First show some with positive exponents: 103 = 2 • What pattern do you notice when you multiply repeatedly by 10? • What pattern do you notice when you divide repeatedly by 10? A1NL11S_c07_0466-0471.indd 467 102 = 467 10 × 10 = 100 10 × 10 × 10 = 1000 Show how each is the number 1 followed by the same number of zeros as the exponent number. Then work backward for 10 to the first power, zero power, and negative powers. With negative exponents, the number of zeros is one less than the exponent number. EX AM P LE Through Cooperative Learning Separate students into groups of three. The first student writes a number in standard form. The second student writes that number in scientific notation, and the third student checks/corrects the work. Have students switch roles so that everyone has done each job at least once. Then repeat, having the first student write a number in scientific notation, the second student write that number in standard form, and the third student check/correct the work. 3 • Why does multiplying by a negative power of 10 result in a smaller number? Lesson 7-2 467 EXAMPLE 4 Additional Examples Example 4 Saturn has a diameter of about 1.2 × 105 km. Its distance from the Sun is about 1,427,000,000 km. from Earth in standard form. 5.91 × 10 8 5.9 1 0 0 0 0 0 0 Move the decimal point 8 places to the right. Standard form refers to the usual way that numbers are written. B. Write Saturn’s distance from the Sun in scientific notation. 1.427 × 109 km 591,000,000 km B Write Jupiter’s average distance from the Sun in scientific notation. ⎩ ⎨ ⎧ 778,400,000 7 7 8, 4 0 0, 0 0 0 Example 5 8 places Order the list of numbers from least to greatest. × × × × £{Î]äääÊ A Write Jupiter’s shortest distance A. Write Saturn’s diameter in standard form. 120,000 km 1.3 2.1 5.4 6.3 Astronomy Application Jupiter has a diameter of about 143,000 km. Its shortest distance from Earth is about 5.91 × 10 8 km, and its average distance from the Sun is about 778,400,000 km. Jupiter’s orbital speed is approximately 1.3 × 10 4 m/s. 7.784 × 10 8 km 10-2, 6.3 × 103, 4.1 × 106, 106, 1 × 10-2, 5.4 × 10-3 10-3, 1 × 10-2, 1.3 × 10-2, 103, 2.1 × 106, 4.1 × 106 Count the number of places you need to move the decimal point to get a number between 1 and 10. Use that number as the exponent of 10. 4a. Use the information above to write Jupiter’s diameter in scientific notation. 1.43 × 10 5 km 4b. Use the information above to write Jupiter’s orbital speed in standard form. 13,000 m/s EXAMPLE 5 Comparing and Ordering Numbers in Scientific Notation Order the list of numbers from least to greatest. 1.2 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5, 1 × 10 -1, 9.9 × 10 -4 INTERVENTION Step 1 List the numbers in order by powers of 10. Questioning Strategies 9.9 × 10 -4, 1.2 × 10 -1, 1 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5 EX A M P L E 4 Step 2 Order the numbers that have the same power of 10. • What does it mean to write a number in standard form? EX A M P L E 9.9 × 10 -4, 1 × 10 -1, 1.2 × 10 -1, 8.2 × 10 4, 2.4 × 10 5, 6.2 × 10 5 5. Order the list of numbers from least to greatest. 5.2 × 10 -3, 3 × 10 14, 4 × 10 -3, 2 × 10 -12, 4.5 × 10 30, 4.5 × 10 14 5 2 × 10 -12, 4 × 10 -3, 5.2 × 10 -3, 3 × 10 14, 4.5 × 10 14, 4.5 × 10 30 • When ordering numbers in scientific notation, why is the power of 10 used to determine the initial order of the numbers? THINK AND DISCUSS 1. Tell why 34.56 × 10 4 is not correctly written in scientific notation. Number Sense Tell students to associate the direction the decimal point moves with the positive and negative directions on a number line: positive numbers to the right and negative numbers to the left. 2. GET ORGANIZED Copy and complete the graphic organizer. *ÜiÀÃÊvÊ£äÊ>`Ê-ViÌvVÊ Ì>Ì Êi}>ÌÛiÊiÝ«iÌ VÀÀiÃ«`ÃÊÌÊÛ}ÊÌ i ¶ `iV>Ê«ÌÊÚÚÚÚÚÚÚÚÚÚ° 468 Chapter 7 Exponents and Polynomials Answers to Think and Discuss 3 Close Summarize 1. 34.56 is not between 1 and 10. A1NL11S_c07_0466-0471.indd 468 Remind students that to multiply by a positive power of 10, they should move the decimal point to the right, and to multiply by a negative power of 10, they should move the decimal point to the left. If a number is in scientific notation, it is in the form x × 10 y, with 1 ≤ x < 10 and with y being any integer. 468 Chapter 7 Ê«ÃÌÛiÊiÝ«iÌ VÀÀiÃ«`ÃÊÌÊÛ}ÊÌ i ¶ `iV>Ê«ÌÊÚÚÚÚÚÚÚÚÚÚ° and INTERVENTION Diagnose Before the Lesson 7-2 Warm Up, TE p. 466 Monitor During the Lesson Check It Out! Exercises, SE pp. 467–468 Questioning Strategies, TE pp. 467–468 Assess After the Lesson 7-2 Lesson Quiz, TE p. 471 Alternative Assessment, TE p. 471 2. See p. A6. 6/25/09 9:09:03 AM 7-2 Exercises 7-2 Exercises KEYWORD: MA7 7-2 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide 1. Vocabulary Explain how you can tell whether a number is written in scientific notation. A number written in sci. notation is a product with 2 parts: a decimal greater than or equal to 1 and less than 10 and a power of 10. SEE EXAMPLE 1 SEE EXAMPLE 2 3 4. 10 -4 0.0001 SEE EXAMPLE 4 p. 468 5 p. 468 5. 10 8 100,000,000 Write each number as a power of 10. 7. 0.000001 10 -6 8. 100,000,000,000,000,000 10 17 Find the value of each expression. 9. 650.3 × 10 6 650,300,000 10. 48.3 × 10 -4 0.00483 p. 467 SEE EXAMPLE 3. 10 -5 0.00001 6. 10,000 10 4 p. 467 SEE EXAMPLE Find the value of each power of 10. 2. 10 6 1,000,000 p. 466 11. 92 × 10 -3 0.092 12. Astronomy A light-year is the distance that light travels in a year and is equivalent to 9.461 × 10 12 km. Write this distance in standard form. 9,461,000,000,000 km 13. Order the list of numbers from least to greatest. 8.5 × 10 -1, 3.6 × 10 8, 5.85 × 10 -3, 2.5 × 10 -1, 8.5 × 10 8 1 2 3 4 5 Extra Practice Skills Practice p. S16 Application Practice p. S34 Find the value of each power of 10. 14. 10 3 1000 15. 10 -9 16. 10 -12 17. 10 14 0.000000001 0.000000000001 100,000,000,000,000 Write each number as a power of 10. 18. 0.01 10 -2 20. 0.000000000000001 10 -15 19. 1,000,000 10 6 Find the value of each expression. 21. 9.2 × 10 4 92,000 22. 1.25 × 10 -7 0.000000125 23. 42 × 10 -5 0.00042 24. 0.05 × 10 7 500,000 25. Biology The human body is made of about 1 × 10 13 cells. Write this number in standard form. 10,000,000,000,000 27. Order the list of numbers from least to greatest. 2.13 × 10 -1, 3.12 × 10 2, 1.23 × 10 -3, 2.13 × 10 1, 1.32 × 10 -3, 3.12 × 10 -3 28. 28. Yes; the smallest grain of pollen is larger than 3 × 10 -7 m. 1.23 × 10 -3, 1.32 × 10 -3, 3.12 × 10 -3, 2.13 × 10 -1, 2.13 × 10 1, 3.12 × 10 2 Health Donnell is allergic to pollen. The diameter of a grain of pollen is between 1.2 × 10 -5 m and 9 × 10 -5 m. Donnell’s air conditioner has a filter that removes particles larger than 3 × 10 -7 m. Will the filter remove pollen? Explain. 29. Entertainment In the United States, a CD is certified platinum if it sells 1,000,000 copies. A CD that has gone 2 times platinum has sold 2,000,000 copies. How many copies has a CD sold if it has gone 27 times platinum? Write your answer in scientific notation. 2.7 × 10 7 Write each number in scientific notation. to indicate the power of 10. For example, to enter 9.2 × 104, press 9.2 4. To enter or powers of 10, use -5 . To enter 10 , press 10 -5 or (-5). Grain of pollen, enlarged 1300 times 1.7 × 10 11 30. 40,080,000 4.008 × 10 7 31. 235,000 2.35 × 10 5 32. 170,000,000,000 33. 0.0000006 6 × 10 -7 35. 0.0412 4.12 × 10 -2 34. 0.000077 7.7 × 10 -5 Number Sense In Exercises 14—17, have students estimate whether their answer is greater than or less than 10 before finding the value. This will help them if they forget the rules for evaluating powers of 10. Technology For Exercises 21—24, students can enter numbers in scientific notation into their calculators by using 26. Statistics At the beginning of the twenty-first century, the population of China was about 1,287,000,000. Write this number in scientific notation. 1.287 × 10 9 7- 2 Powers of 10 and Scientific Notation A1NL11S_c07_0466-0471.indd 469 If you finished Examples 1–5 Basic 14—46, 48—52, 55—63 Average 14—53, 55—63 Advanced 14—45, 47—63 Quickly check key concepts. Exercises: 16, 18, 22, 26, 27, 36 PRACTICE AND PROBLEM SOLVING 14–17 18–20 21–24 25–26 27 If you finished Examples 1–3 Basic 14—24 Average 14—24, 53 Advanced 14—24, 53 Homework Quick Check 5.85 × 10 -3, 2.5 × 10 -1, 8.5 × 10 -1, 3.6 × 10 8, 8.5 × 10 8 Independent Practice For See Exercises Example Assign Guided Practice exercises as necessary. 469 7/18/09 4:50:48 PM KEYWORD: MA7 Resources Lesson 7-2 469 State whether each number is written in scientific notation. If not, write it in scientific notation. Chemistry 40. 0.1 41. 7 × 10 8 42. 48,000 43. 3.5 × 10 -6 45. Communication This bar graph shows the increase of cellular telephone subscribers worldwide. a. Write the number of subscribers for the following years in standard form: 1999, 2000, and 2003. 40. no; 1 × 45a. 490,000,000; 740,000,000; 1,329,000,000 41. yes 42. no; 4.8 × 104 43. yes 49b. Possible answer: It would be easy to accidentally omit a 0 or add an extra 0 when writing the number in standard form. You are probably less likely to make an error when using scientific notation. b. When you double 7.4 × 10 8, you get approx. 14 × 10 8, or 1.4 × 10 9 in sci. notation. 1.4 is close to 1.3, so Zorah’s observation is correct. ÈÊÊÊ£äÊnÊ {ÊÊÊ£äÊnÊ ÓÊÊÊ£äÊnÊ £°ÎÓÊÊÊ£äÊ nÊÊÊ£äÊnÊ £°£xxÊÊÊ£äÊ £°{ÊÊÊ£äÊÊ £°ÓÊÊÊ£äÊÊ £ÊÊÊ£äÊÊ °xÊÊÊ£äÊn b. Zorah looks at the bar graph and says, “It looks like the number of cell phone subscribers nearly doubled from 2000 to 2003.” Do you agree with Zorah? Use scientific notation to explain your answer. 39. no; 2.5 × 102 10-1 7À`Ü`iÊ iÊ* iÊ-ÕLÃVÀLiÀÃ Ç°{ÊÊÊ£äÊn 38. no; 1.2 × 106 39. 0.25 × 10 3 {°ÊÊÊ£äÊn 37. yes 38. 1,200,000 electron The image above is a colored bubble-chamber photograph. It shows the tracks left by subatomic particles in a particle accelerator. 36. no; 5 × 10-4 37. 8.1× 10 -2 44. Chemistry Atoms are made of three elementary particles: protons, electrons, and neutrons. The mass of a proton is about 1.67 × 10 -27 kg. The mass of an electron is about 0.000000000000000000000000000000911 kg. The mass of a neutron is about 1.68 × 10 -27 kg. Which particle has the least mass? (Hint: Compare the numbers after they are written in scientific notation.) Reading Math For Exercise 44, in 1.67 × 10 -27, 1.67 is called the coefficient. Answers 36. 50 × 10 -5 -ÕLÃVÀLiÀÃ Exercise 49 involves writing numbers in scientific notation. This exercise prepares students for the Multi-Step Test Prep on page 494. £ Óäää Óää£ ÓääÓ ÓääÎ 9i>À 46. Measurement In the metric system, the basic unit for measuring length is the meter (m). Other units for measuring length are based on the meter and powers of 10, as shown in the table. b. 10 -3 = 0.001; 10 -2 = 0.01; 10 -1 = 0.1; 10 1 = 10; 10 2 = 100; 10 3 = 1000 Selected Metric Lengths 1 millimeter (mm) = 10 -3 m 1 centimeter (cm) = 10 1 decimeter (dm) = 10 -2 -1 1 dekameter (dam) = 10 1 m m 1 hectometer (hm) = 10 2 m m 1 kilometer (km) = 10 3 m a. Which lengths in the table are longer than a meter? Which are shorter than a meter? How do you know? dam, hm, km; mm, cm, dm b. Evaluate each power of 10 in the table to check your answers to part a. 1 = 10 -3. Based on this information, 47. Critical Thinking Recall that ___ 10 3 complete the following statement: Dividing a number by 10 3 is equivalent to multiplying by . 10 -3 48. Write About It When you change a number from scientific notation to standard form, explain how you know which way to move the decimal point and how many places to move it. If the exp. is pos., move the dec. pt. that many places to the right. If the exp. is neg., move the dec. pt. that many places to the left. 49. This problem will prepare you for the Multi-Step Test Prep on page 494. 8 a. The speed of light is approximately 3 × 10 m/s. Write this number in standard form. 300,000,000 b. Why do you think it would be better to express this number in scientific notation rather than standard form? c. The wavelength of a shade of red light is 0.00000068 meters. Write this number in scientific notation. 6.8 × 10 -7 7-2 PRACTICE A 7-2 PRACTICE C ________________________________________ LESSON 7-2 __________________ __________________ Practice B 7-2 PRACTICE B Powers of 10 and Scientific Notation 470 Find the value of each power of 10. 1. 10−3 0.001 2. 105 100,000 3. 10−4 4. 100 1 5. 107 10,000,000 6. 101 LESSON 10 7-2 Write each number as a power of 10. 7. 1,000,000 10 10 10. 0.00001 6 −5 __________________ __________________ Reading Strategies 7-2 READING STRATEGIES Use a Graphic Aid The graphic aid below summarizes how to work with powers of 10. 8. 0.001 10 −3 10 −1 11. 0.1 Chapter 7 Exponents and Polynomials ________________________________________ 0.0001 12. 0.00000001 10 5020 14. 603 × 10−4 0.0603 15. 52.8 × 106 52,800,000 16. 5.41 × 10−3 0.00541 17. 0.03 × 10−2 0.0003 18. 22.81 × 10−6 19. 4500 4.5 × 10 21. 0.00002 2 × 10−5 Find the value of 105. Find the value of 10 −4. Step 1: Start with the number 1. Step 1: Start with the number 1. 1.0 Step 2: The exponent is positive 5. Move the decimal 5 spaces to the right. 22. 0.00203 Step 2: The exponent is negative 4. Move the decimal 4 spaces to the left. 1.0 0 0 0 0 = 100,000 0.00002281 0 . 0 0 1 = 0.0001 Write 100,000,000 as a power of 10. 20. 6,560,000 __________________ 7-2 RETEACH Powers of 10 and Scientific Notation 1.0 A1NL11S_c07_0466-0471.indd 470 Write each number in scientific notation. 3 __________________ Review for Mastery The exponent will tell you how many places to move the decimal when finding the value of a power of 10. −8 Find the value of each expression. 13. 5.02 × 103 7-2 Powers of 10 are used to write large numbers in a simple way. 10−6 9. 0.000001 ________________________________________ LESSON Write 0.00001 as a power of 10. 1 0 0 0 0 0 0 0 0. 6.56 × 106 0.0 0 0 0 1 The decimal point is 8 places to the right of the 1. The exponent is 8. 2.03 × 10−3 100,000,000 = 10 Order the list of numbers from least to greatest. 23. 3 × 102 ; 4.54 × 10−3 ; 6.75 × 102 ; 8.2 × 10−4 ; 9 × 10−1 ; 6.18 × 10−4 The decimal point is 5 places to the left of the one. The exponent is −5. 0.00001 = 10−5 8 Numbers greater than 1 will have a positive exponent. 6.18 × 10−4; 8.2 × 10−4; 4.54 × 10−3; 9 × 10−1; 3 × 102; 6.75 × 102 Numbers less than 1 will have a negative exponent. 24. 5.4 × 10−3 ; 6.2 × 10−1 ; 7.25 × 103 ; 6.87 × 103 ; 2.24 × 10−1 ; 6.6 × 10−3 5.4 × 10−3; 6.6 × 10−3; 2.24 × 10−1; 6.2 × 10−1; 6.87 × 103; 7.25 × 103 25. In 1970, the number of televisions sold in the United States was about 1.2 × 107. Write this number in standard form. 26. In 1950, about 3,880,000 households in the United States had televisions. Write this number in scientific notation. 27. Find the volume of the cube shown at right. Write the answer in both standard form and in scientific notation. 64,000,000,000 mm3 6.4 × 1010 mm3 First determine whether the decimal point will move to the right or to the left. Then find the value of each power of 10. Complete each of the following. 12,000,000 3.88 × 106 1. 106 1. Which represents a very large number: 4 × 109 or 4 × 10−9 ? 4 × 109 2. Write 9.7 × 10−3 in standard form. 0.0097 4.19 × 1011 3. Write 419,000,000,000 in scientific notation. Match each number with its equivalent power of 10. A. 104 B. 10 −5 C. 105 4. 0.00001 B 5. 10,000 A 6. 100,000 C 7. 0.0001 D 470 Chapter 7 56,000,000 9. 87.5 × 104 875,000 2. 10−2 3. 104 left 0.01 right 10,000 First determine whether the exponent will be positive or negative when each number is written as a power of 10. Then write each number as a power of 10. 4. 1000 D. 10 −4 Find the value of each expression. 8. 56 × 106 right 1,000,000 positive 103 5. 0.0001 negative 10−4 6. 10,000,000 positive 107 7/18/09 4:50:53 PM If students chose A or D in Exercise 50, they may have associated the exponent of 10 with the number of zeros in the answer. 50. There are about 3.2 × 10 7 seconds in one year. What is this number in standard form? 0.000000032 0.00000032 In Exercise 52, encourage students to write the second and fourth numbers in the list in scientific notation as a first step. 32,000,000 320,000,000 51. Which expression is the scientific notation for 82.35? 8.235 × 10 1 823.5 × 10 -1 8.235 × 10 -1 0.8235 × 10 2 52. Which statement is correct for the list of numbers below? 2.35 × 10 -8, 0.000000029, 1.82 × 10 8, 1,290,000,000, 1.05 × 10 9 The list is in increasing order. If 0.000000029 is removed, the list will be in increasing order. If 1,290,000,000 is removed, the list will be in increasing order. The list is in decreasing order. CHALLENGE AND EXTEND 53. About 7 times; 53. Technology The table shows estimates of Computer Storage computer storage. A CD-ROM holds 700 MB. 4.7 GB, the storage 1 kilobyte (KB) ≈ 1000 bytes A DVD-ROM holds 4.7 GB. Estimate how many of the DVD, is the 1 megabyte (MB) ≈ 1 million bytes times more storage a DVD has than a CD. same as 4700 MB, Explain how you found your answer. 1 gigabyte (GB) ≈ 1 billion bytes which is approx. 7 times 700 MB, the 54. For parts a–d, use what you know about multiplying by powers of 10 and the storage of the CD. Commutative and Associative Properties of Multiplication to find each product. Write each answer in scientific notation. 3 a. (3 × 10 2)(2 × 10 ) 6 × 10 5 b. (5 × 10 8)(1.5 × 10 -6) 7.5 × 10 2 c. (2.2 × 10 -8)(4 × 10 -3) 8.8 × 10 -11 d. (2.5 × 10 -12)(2 × 10 6) 5 × 10 -6 e. Based on your answers to parts a–d, write a rule for multiplying numbers in scientific notation. f. Does your rule work when you multiply (6 × 10 3)(8 × 10 5)? Explain. 54e. First multiply the numbers, and then multiply the powers of 10 by adding the exponents. Yes, but the answer, 48 × 10 8, is not in sci. notation. After multiplying, you will have to rewrite the answer in sci. notation as 4.8 × 10 9. SPIRAL REVIEW Define a variable and write an inequality for each situation. Graph the solutions. (Lesson 3-1) Technology In Exercise 53, you might tell students that metric prefixes are used to describe computer storage. However, bytes do not follow true metric conventions. For example, 1 kilobyte = 1024 bytes, not 1000 bytes. Answers 55–57. For graphs, see p. A27. Journal Have students explain why 105 has 5 zeros, but 10-5 has only 4 zeros. Have students write four numbers in scientific notation, two with positive exponents and two with negative exponents, and then arrange them from least to greatest. 55. Let m = number 55. Melanie must wait at least 45 minutes for the results of her test. of minutes; m ≥ 45. 56. Ulee’s dog can lose no more than 8 pounds to stay within a healthy weight range. 7-2 56. Let p = pounds; 57. Charlene must spend more than $50 to get the advertised discount. p ≤ 8 where p is nonneg. Solve each system by elimination. (Lesson 6-3) 57. Let m = money 58. ⎧⎨ x + y = 8 (5, 3) ⎩x-y=2 spent; m > 50. 59. ⎧ 2x + y = -3 (-2, 1) ⎧ x - 6y = -3 60. ⎨ ⎨ ⎩ 2x + 3y = -1 ⎩ 3x + 4y = 13 Find the value of each expression. (3, 1) Evaluate each expression for the given value(s) of the variable(s). (Lesson 7-1) _ 61. t -4 for t = 2 1 16 62. (-8m)0 for m = -5 1 LESSON 7-2 7-2 PROBLEM SOLVING Problem Solving A1NL11S_c07_0466-0471.indd 1. Insects can multiply rapidly during the summer. A pair of houseflies could potentially grow to a population of 1.91 1020. If all the descendants of a female cabbage aphid lived, the population could increase to 1.56 1024. 471 Which population would be larger? Name _______________________________________ Date __________________ Class__________________ LESSON 7-2 Powers of 10 and Scientific Notation Write the correct answer 2. The graph shows the gross domestic product (GDP) for several countries around the world. Identify the country whose GDP is twice that of another country. Write the GDPs of both countries in standard form. 3.38 107 Philippines United Kingdom 8.79 10 6.04 107 1. C B 3.8 101 AU D 38 10 2 AU 3. 8 10 1 AU 7. What is the diameter of the Earth in scientific notation? F 1.28 102 km H 1.28 10 4 km G 1.28 103 km J 1.28 105 km about 5.2 times a. How many times farther from the about 39.5 times sun is Pluto than Earth? b. How many times farther from the about 7.6 times sun is Pluto than Jupiter? 5. Suppose the mass of Mars were written in standard form. How many digits would be to the left of the decimal? F 23 G B 1.50 109 km D 1.50 1011 km A 0.38 AU 4.84 108 miles is Jupiter than Earth? 1.50 108 km C 1.50 1010 km 6. Which of these is the average distance from the Sun to Mercury expressed in scientific notation? 9.3 107 miles the sun in scientific notation. List the countries in order of population size from least to greatest. A a. Write the distance between Earth and 2. The average distance between Pluto and the sun is 3,675,000,000 miles. 4. An AU is an astronomical unit. One AU equals 150,000,000 km. What is that measure in scientific notation? H 25 24 3. The star closest to the sun is 25,000,000,000,000 miles from the sun. How many times farther is it from the sun to the nearest star as it is from Earth to the sun? 4. How many miles does light travel in a year? Give your 5.9 1012 answer in scientific notation rounded to the nearest tenth. 5. a. One galaxy is 200,000 light-years from the sun. J 26 b. How many times farther is this galaxy from the sun than Earth is from the sun? Planet about 269,000 times One light-year is the distance light travels in one year. Light travels 186,282 miles in 1 second. about 1.18 1018 How many miles from the sun is the galaxy? Astronomical Data for the First Five Planets Avg. Distance Diameter Mass (kg) from Sun (AU) (km) Mercury 0.38 4,880 3.20 1023 Venus 0.72 12,100 4.87 1024 Earth 1 12,800 5.97 1024 Mars 1.52 6,790 6.42 1023 Jupiter 5.20 143,000 1.90 1027 about 1.3 1010 times You can use scientific notation to compare masses of large objects with masses of small objects. Mass of hydrogen atom: 1.67 10 24 grams Jerry’s mass: Mass of Earth: Holt McDougal Algebra 1 7/18/09 4:51:04 PM a. Write the area of the Pacific Ocean in standard form. 64,000,000 mi2 b. Write the volume of the Pacific Ocean in scientific notation. 1.7 × 108 mi3 4. Order the list of numbers from least to greatest. 3.6 × 10 -3, 1 × 10 -5, 2.7 × 102, 1.3 × 104, 3.1 × 104, 4.1 × 10-3 1 × 10 -5, 3.6 × 10 -3, 4.1 × 10 -3, 2.7 × 102, 1.3 × 104, 3.1 × 104 6. Write a ratio to compare Jerry’s mass with that of a hydrogen atom. about 3.8 1028 to 1 about 3.6 1051 to 1 7-17 471 Also available on transparency 6.35 101 kilograms 5.97 1024 kilograms 7. Write a ratio to compare the mass of Earth with that of a hydrogen atom. Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 3. The Pacific Ocean has an area of about 6.4 × 10 7 square miles. Its volume is about 170,000,000 cubic miles. Answer the following questions. 7 Ethiopia: $54,000,000,000; Cambodia: $27,000,000,000 125 How many times farther from the sun is Jupiter than Earth? c. How many times farther from the Sun Kenya, UK, Philippines, Brazil, India 0.00293 7-2 CHALLENGE and the sun in scientific notation. The table shows astronomical data about several planets. Use the table to answer questions 4–7. Select the best answer. 2. 29.3 × 10-4 Using Scientific Notation to Make Comparisons b. Write the distance between Jupiter 3. The 2005 population estimates of five countries are listed below. Brazil 1.86 108 India 1.08 109 3 _ The average distance between Earth and the sun is 93,000,000 miles. The average distance between Jupiter and the sun is 484,000,000 miles. Using scientific notation, you can answer questions like the one below. cabbage aphid Kenya Challenge 3,745,000 63. 3a -3b 0 for a = 5 and b = 6 7- 2 Powers of 10 and Scientific Notation Name _______________________________________ Date __________________ Class__________________ 1. 37.45 × 105 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-16 Holt McDougal Algebra 1 Lesson 7-2 471 7-3 Organizer Explore Properties of Exponents Use with Lesson 7-3 Pacing: You can use patterns to find some properties of exponents. Traditional 1 day 1 Block __ day 2 Objective: Use patterns to Use with Lesson 7-3 GI discover multiplication properties of exponents. <D @<I Activity 1 Online Edition 1 Copy and complete the table below. 3 2 3 3 = (3 · 3)(3 · 3 · 3) = 3 5 Countdown Week 15 54 52 = ( Resources Algebra Lab Activities 5 )( 5 5 )= 56 4 )= 4 5 5 43 43 = ( 4 4 4 )( 4 4 23 22 = ( 2 2 2 )( 2 2 )= 2 63 64 = ( 7-3 Lab Recording Sheet 5 )( 6 5 ) = 6 · 6 · 6; 6 · 6 · 6 · 6; 6 7 Teach 2 Examine your completed table. Look at the two exponents in each factor and the exponent in the final answer. What pattern do you notice? Discuss 3 Use your pattern to make a conjecture: a m · a n = a The exp. in the final answer is the sum of the exponents in the orig. problem. . m+n Remind students that a conjecture is an educated guess that is based on evidence but has not been proven true or false. Try This Use your conjecture to write each product below as a single power. 2. 7 2 · 7 2 7 4 3. 10 8 · 10 4 10 12 1. 5 3 · 5 5 5 8 Discuss with students what patterns they notice in each activity. Help students represent the patterns in conjectures with words and with algebraic statements. For example, Activity 3 can be thought of as “distributing” the exponent. 4. 8 7 · 8 3 8 10 5. Make a table similar to the one above to explore what happens when you multiply more than two powers that have the same base. Then write a conjecture in words to summarize what you find. Activity 2 1 Copy and complete the table below. (2 3)2 = 2 3 2 3 = ( 2 2 (2 2)3 = 2 2 2 2 2 2 = ( 2 (4 ) 2 4 = 2 )( 2 2 )( 2 42 42 42 42 = ( 4 (3 4)2 = 3 4 3 4 = ( 3 3 3 2 2 )=26 2 )( 2 2 )= 2 6 4 )( 4 4 )( 4 4 )( 4 3 )( 3 3 3 )= 3 3 4 )= 4 8 8 (6 3)4 = 6 3 · 6 3 · 6 3 · 6 3 = (6 · 6 · 6)(6 · 6 · 6)(6 · 6 · 6)(6 · 6 · 6) = 6 12 2 Examine your completed table. Look at the two exponents in the original expression and the exponent in the final answer. What pattern do you notice? The exp. in the final answer is the product nof the exponents in the orig. problem. 3 Use your pattern to make a conjecture: (a m) = a . mn 472 Chapter 7 Exponents and Polynomials Answers to Try This 5. Possible answer: 3 · 33 · 34 = (3 · 3) · (3 · 3 · 3) · (3 · 3 · 3 · 3) = 39 A1NL11S_c07_0472-0473.indd 2472 22 · 22 · 23 = (2 · 2) · (2 · 2) · (2 · 2 · 2) = 27 42 · 45 · 43 = (4 · 4) · (4 · 4 · 4 · 4 · 4) · (4 · 4 · 4) = 410 62 · 63 · 63 · 62 = (6 · 6) · (6 · 6 · 6) · (6 · 6 · 6) · (6 · 6) = 610 When multiplying powers with the same base, the answer is the base raised to the sum of all the exponents. KEYWORD: MA7 Resources 472 Chapter 7 6/25/09 9:08:21 AM Close Try This Key Concepts Use your conjecture to write each product below as a single power. 4 2 2 6. (5 3) 5 6 7. (7 2) 7 4 8. (3 3) 3 12 The product of two powers with the same base is the base raised to the sum of the exponents. (9 ) 9 21 7 3 9. 10. Make a table similar to the one in Activity 2 to explore what happens when you raise a power to two powers, for example, ⎡⎣(4 2) words to summarize what you find. ⎤⎦ . Then write a conjecture in 3 3 The power of a power is the base raised to the product of the exponents. The power of a product is the product of each factor raised to that exponent. Activity 3 1 Copy and complete the table below. Assessment (ab)3 = (ab)(ab)(ab) = (a a a)(b b b) = a 3 b (mn)4 = (mn)(mn)(mn)(mn) = ( m m m m )( n (xy)2 = ( xy )( xy ) = ( x x )( y y ) = x 2 y Journal Have students use numbers to show an example of each property explored. 3 n n n )= m 4 n 4 2 (cd)5 = (cd )(cd )(cd )(cd )(cd ) = ( c c c c c )( d d d d d ) = c 5 d 5 (pq)6 = pq · pq · pq · pq · pq · pq ; (p · p · p · p · p · p) ·(q · q · q · q · q · q) ; p 6q 6 2 Examine your completed table. Look at the original expression and the final answer. What pattern do you notice? To get the final answer, the exp. is “distributed” to each factor in the product. n b . n; n 3 Use your pattern to make a conjecture: (ab) = a Try This Use your conjecture to write each power below as a product. 7 11. (rs)8 r 8s 8 12. (yz)9 y 9z 9 13. (ab) a 7b 7 14. (xz)12 x 12z 12 15. Look at the first row of your table. What property or properties allow you to write (ab)(ab)(ab) as (a · a · a)(b · b · b)? Assoc. and Comm. Properties of Mult. 16. Make a table similar to the one above to explore what happens when you raise a product containing more than two factors to a power, for example, (xyz)7. Then write a conjecture in words to summarize what you find. Possible answer: (abc)3 = (abc)(abc)(abc) = (a · a · a)(b · b · b)(c · c · c) = a 3b 3c 3 (xyz)2 = (xyz)(xyz) = (x · x)(y · y)(z · z) = x 2y 2z 2 (mnpq)4 = (mnpq)(mnpq)(mnpq)(mnpq) = (m · m · m · m)(n · n · n · n)(p · p · p · p)(q · q · q · q) = m 4 n 4 p 4 q 4 When a product is raised to a power, the exp. is “distributed” to each factor in the product. 7- 3 Algebra Lab 473 Answers to Try This 10. Possible answer: A1NL11S_c07_0472-0473.indd 473 [(3 ) ] [(2 ) ] [(4 ) ] 4 2 3 = (32) · (32) · (32) · (32) = 36 · 36 · 36 · 36 = 324 3 2 2 = (22) · (22) · (22) = 24 · 24 · 24 = 212 3 2 5 = (42) · (42) · (42) = 410 · 410 · 410 = 430 3 2 5 3 2 5 3 3 6/25/09 9:08:28 AM 2 5 When a power is raised to 2 powers, the final answer is the base raised to the product of all the exponents. 7-3 Algebra Lab 473 7-3 Organizer Pacing: Traditional 1 day Block Multiplication Properties of Exponents 7-3 __1 day 2 Objective: Use multiplication GI properties of exponents to evaluate and simplify expressions. <D @<I Online Edition Who uses this? Astronomers can multiply expressions with exponents to find the distance between objects in space. (See Example 2.) Objective Use multiplication properties of exponents to evaluate and simplify expressions. Tutorial Videos You have seen that exponential expressions are useful when writing very small or very large numbers. To perform operations on these numbers, you can use properties of exponents. You can also use these properties to simplify your answer. Countdown Week 15 In this lesson, you will learn some properties that will help you simplify exponential expressions containing multiplication. Warm Up Simplifying Exponential Expressions Write each expression using an exponent. 1. 2 · 2 · 2 An exponential expression is completely simplified if… 23 • There are no negative exponents. 2. x · x · x · x 1 1 3. _ 4-2 or _2 4·4 4 Write each expression without using an exponent. x4 4·4·4 4. 43 6. m -4 5. y 2 • The same base does not appear more than once in a product or quotient. • No powers are raised to powers. • No products are raised to powers. • No quotients are raised to powers. • Numerical coefficients in a quotient do not have any common factor other than 1. y·y 1 __ m·m·m·m Examples b x3 _ a Also available on transparency z 12 4 a b 4 Nonexamples 5a 2 s5 _ _ t 5 2b a -2 ba x · x 2 (z 3)4 (ab)4 (_st ) 5 10a 2 _ 4b Products of powers with the same base can be found by writing each power as repeated multiplication. Q: What do you call xsun? Notice the relationship between the exponents in the factors and the exponent in the product: 5 + 2 = 7. A: Solar power. Product of Powers Property WORDS NUMBERS The product of two powers with the same base equals that base raised to the sum of the exponents. 474 67 · 6 4 = 6 7+4 = 6 11 ALGEBRA If a is any nonzero real number and m and n are integers, then a m · a n = a m+n. Chapter 7 Exponents and Polynomials 1 Introduce A11NLS_c07_0474-0480.indd e x p l o474 r at i o n 7-3 Multiplication Properties of Exponents 1. The expression x · x · x · x · x can be evaluated two ways. As a product of two powers, it can be written as x 3 · x 2. As a single power, it can be written as x 5. Use this information to complete the table below. Expression Product of Powers Single Power x x x x x x3 x2 x5 y y y y y y a a a a a a a a a m m m m m m 2. Describe any patterns you see in the table above. 3. Use a similar method to complete the table below. Expression Product of Powers Single Power y y y y y y y y3 y2 y2 y7 b b b b b b b b b KEYWORD: MA7 Resources z z z z z z z z z z x x x x x x x x THINK AND DISCUSS 4. Describe any patterns you see in the second table. 474 Chapter 7 5. Explain how you can use your findings to write x 10 · x 4 as a single power. Motivate Draw a square on the board. Label one side x3. Ask students for an expression for the area of the square. x3 · x3 Show students that this can also be written as (x3)2, and tell them that the properties in this lesson will show them how to simplify expressions with multiple exponents, such as this one. Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 8/18/09 9:54:52 AM EXAMPLE 1 Finding Products of Powers Simplify. Additional Examples A 25 · 26 2 ·2 2 5+ 6 2 11 5 6 2 -2 2 -2 B 4 ·3 C a ·b ·a 5 5 Simplify. A. 32 · 35 ·4 ·3 4 · 3 · 45 · 36 (4 2 · 4 5) · (3 -2 · 3 6) 4 2+5 · 3 -2+ 6 47 · 34 4 Example 1 Since the powers have the same base, keep the base and add the exponents. 6 Group powers with the same base together. B. 24 · 34 · 2-2 · 32 Add the exponents of powers with the same base. C. q3 · r2 · q6 q9r2 D. n3 · n-4 · n 1 Example 2 Group powers with the same base together. Add the exponents of powers with the same base. D y 2 · y · y -4 (y 2 · y 1) · y -4 Group the first two powers. y 3 · y -4 The first two powers have the same base, so add the exponents. The two remaining powers have the same base, so add the exponents. y -1 1 _ y 1b. 3 -3 · 5 8 · 3 4 · 5 2 3 × 5 10 1d. x · x -1 · x -3 · x -4 1 _ n4 2 Light from the Sun travels at about 1.86 × 105 miles per second. It takes about 15,000 seconds for the light to reach Neptune. Find the approximate distance from the Sun to Neptune. Write your answer in scientific notation. 2.79 × 109 mi Write with a positive exponent. Simplify. 1a. 7 8 · 7 4 7 12 m5 1c. m · n -4 · m 4 EXAMPLE 22 · 36 2 a4 · b5 · a2 (a 4 · a 2)· b 5 a6 · b5 a 6b 5 A number or variable written without an exponent actually has an exponent of 1. 10 = 10 1 y = y1 37 _ INTERVENTION x7 Questioning Strategies EX AM P LE Astronomy Application • How do you know when an expression containing exponents is completely simplified? Light from the Sun travels at about 1.86 × 10 5 miles per second. It takes about 500 seconds for the light to reach Earth. Find the approximate distance from the Sun to Earth. Write your answer in scientific notation. • How do you know which powers to group together? distance = rate × time = (1.86 × 10 5) × 500 = (1.86 × 10 5) × (5 × 10 2) 1 Write 500 in scientific notation. EX AM P LE 2 • What is the formula for distance? = (1.86 × 5) × (10 × 10 5 ) 2 = 9.3 × 10 7 Use the Commutative and Associative Properties to group. • Why should you write the time in scientific notation? • If you left the time in standard notation, how would that affect your calculations? Multiply within each group. The Sun is about 9.3 × 10 miles from Earth. 7 2. Light travels at about 1.86 × 10 5 miles per second. Find the approximate distance that light travels in one hour. Write your answer in scientific notation. 6.696 × 10 8 mi 7-3 Multiplication Properties of Exponents 475 2 Teach A1NL11S_c07_0474-0480.indd 475 6/25/09 9:11:42 AM Guided Instruction Introduce the Product of Powers Property by writing each power in the product in factored form and having students discover the relationship among the exponents. Have students discover the other properties in a similar fashion. In Example 1, encourage students to add the exponents of powers with the same base before rewriting negative exponents as positive exponents. Through Graphic Organizers Have students create a graphic organizer to show evidence that each property works. Product of Powers Power of a Power Power of a Product 32 · 33 9 · 27 243 (23)4 84 4096 (2 · 4)3 83 512 35 243 212 4096 23 · 43 8 · 64 512 Lesson 7-3 475 To find a power of a power, you can use the meaning of exponents. Additional Examples Example 3 Notice the relationship between the exponents in the original power and the exponent in the final power: 3 · 2 = 6. Simplify. A. (52) 58 B. ( ) 1 C. ( ) · x4 4 0 43 -5 x3 Power of a Power Property WORDS 1 _ x11 NUMBERS A power raised to another power equals that base raised to the product of the exponents. INTERVENTION EXAMPLE Questioning Strategies 3 ALGEBRA If a is any nonzero real number and m and n are n integers, then (a m) = a mn. (6 7) 4 = 6 7 · 4 = 6 28 Finding Powers of Powers Simplify. EX A M P L E 3 A (7 4) 3 74·3 7 12 • What operation is performed on the exponents when a power is raised to another power? Use the Power of a Power Property. Simplify. B (3 ) 6 0 36·0 30 1 Reading Math Tell students that the exponent is usually read as an ordinal followed by the word power. For example, 74 is read “seven to the fourth power.” Point out that 2 and 3 are exceptions to this rule. For those exponents, it is customary to say “squared” or “cubed,” respectively. If students have trouble with ordinals, suggest that they make a chart with the numbers in the first ENGLISH column and their ordinals LANGUAGE LEARNERS in the second column. Use the Power of a Power Property. Zero multiplied by any number is zero. Any number raised to the zero power is 1. C (x ) 2 -4 ·x 5 x 2 · (-4) · x 5 x -8 x ·x Use the Power of a Power Property. 5 Simplify the exponent of the first term. -8 + 5 Since the powers have the same base, add the exponents. x -3 1 _ x3 Write with a positive exponent. Simplify. 3a. (3 4) 5 3 20 3b. (6 0) 3 3c. (a 3) · (a -2) 4 1 -3 a 18 Multiplication Properties of Exponents Sometimes I can’t remember when to add exponents and when to multiply them. When this happens, I write everything in expanded form. Briana Tyler Memorial High School 476 Chapter 7 3 Then (x 2) = x 2 · 3 = x 6. 3 This way I get the right answer even if I forget the properties. Chapter 7 Exponents and Polynomials A11NLS_c07_0474-0480.indd 476 476 For example, I would write x 2 · x 3 as (x · x)(x · x · x) = x 5. Then x 2 · x 3 = x 2 + 3 = x 5. I would write (x 2) as x 2 · x 2 · x 2, which is (x · x)(x · x)(x · x) = x 6. 12/11/09 9:25:56 PM Powers of products can be found by using the meaning of an exponent. "" Ê,,", ,/ Students sometimes add exponents when a power is raised to another power. Encourage students to write out the expression in expanded form to help them remember. Power of a Product Property WORDS NUMBERS (2 · 4) = 2 · 4 3 A product raised to a power equals the product of each factor raised to that power. EXAMPLE 4 3 ALGEBRA If a and b are any nonzero real numbers and n is any n integer, then (ab) = a nb n. 3 = 8 · 64 = 512 Communicating Math Have students write the Product of Powers, Power of a Power, and Power of a Product Properties in their own words. Then have volunteers read their definitions aloud. Finding Powers of Products Simplify. A (-3x)2 (-3)2 · x 2 Use the Power of a Product Property. 9x 2 Simplify. Additional Examples B - (3x)2 In Example 4B, the negative sign is not part of the base. -(3x)2 = -1 · (3x)2 - (3 2 · x 2) - (9 · x 2) -9x 2 C x ·y x -6 · y 0 x -6 · 1 1 _ x6 Simplify. Simplify. (x -2 · y 0)3 (x -2)3 · (y 0)3 -2 · 3 Example 4 Use the Power of a Product Property. 2 B. (-2y) 3 -4y2 -8y3 2 x12 _ C. (x6 · y-3) y6 Use the Power of a Product Property. 0·3 A. -(2y) Use the Power of a Power Property. Simplify. Write y 0 as 1. Write with a positive exponent. INTERVENTION Questioning Strategies Simplify. 4a. (4p)3 64p 3 4b. (-5t ) 25t 2 2 4 4c. (x y ) · (x y ) 2 3 4 2 4 -4 EX AM P LE 1 _ • How does the Power of a Product Property compare with the Distributive Property? y4 THINK AND DISCUSS 1. Explain why (a ) 2 3 4 • If a negative number is raised to an even-numbered power, is it positive or negative? Why? and a 2 · a 3 are not equivalent expressions. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, supply the missing exponents. Then give an example for each property. ÕÌ«V>ÌÊ*À«iÀÌiÃÊvÊÝ«iÌÃ *À`ÕVÌÊvÊ *ÜiÀÃÊ*À«iÀÌÞ *ÜiÀÊvÊ> *ÜiÀÊ*À«iÀÌÞ *ÜiÀÊvÊ> *À`ÕVÌÊ*À«iÀÌÞ Ê>ÊÊˁ >ÊÊÊÊ>Ê Ê Ê> Ê ÊÊ®ÊÊÊÊ>Ê Ê Ê>L®ÊÊÊÊ>Ê ÊLÊ Ê 7-3 Multiplication Properties of Exponents Answers to Think and Discuss 3 Close A11NLS_c07_0474-0480.indd 477 Summarize Have students identify the property they can use to simplify each expression. Then have them simplify the expression. (32)3 Power of a Power; 36 x2 · x4 Product of Powers; x6 16b2 Power of a Product; _ c4 (4b · c-2)2 477 1. (a2) = a2·3 = a6, while a2 · AM a3 = a2+3 = a5. 8/18/09 10:13:30 3 and INTERVENTION 2. See p. A6. Diagnose Before the Lesson 7-3 Warm Up, TE p. 474 Monitor During the Lesson Check It Out! Exercises, SE pp. 475–477 Questioning Strategies, TE pp. 475–477 Assess After the Lesson 7-3 Lesson Quiz, TE p. 480 Alternative Assessment, TE p. 480 Lesson 7-3 477 7-3 7-3 Exercises Exercises KEYWORD: MA7 7-3 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide SEE EXAMPLE Assign Guided Practice exercises as necessary. 2. 5 3 · 5 3 5 6 Simplify. SEE EXAMPLE 6. 3 p. 476 9. SEE EXAMPLE 4 p. 477 4. x 2 · x -3 · x 4 x 3 3. n 6 · n 2 n 8 5. Science If you traveled in space at a speed of 1000 miles per hour, how far would you travel in 7.5 × 10 5 hours? Write your answer in scientific notation. 7.5 × 10 8 mi 2 p. 475 If you finished Examples 1–4 Basic 18—65, 68, 75, 80—83, 98—106 Average 18—43, 44—52 even, 53, 54, 56—64 even, 66—83, 84—96 even, 98—106 Advanced 18—42 even, 44—54, 56—106 Quickly check key concepts. Exercises: 18, 22, 28, 32, 40, 42, 46 Simplify. 1. 2 2 · 2 3 2 5 SEE EXAMPLE If you finished Examples 1–2 Basic 18—22, 53, 68 Average 18—22, 53, 68, 84 Advanced 18—22, 53, 68, 84 Homework Quick Check 1 p. 475 (x 2) 5 x 10 (3 -2) 2 1 , or 1 4 _ _ 10. 81 3 12. (2t)5 32t 5 15. 7. (-2x ) -8x 15 5 3 (y 4) 8 y 32 (a ) -3 4 · (a (p 3) 3 8. p9 11. xy · (x 2) · (y 3) ) a2 3 7 2 4 x 7y 13 2 13. (6k) 36k 2 14. (r 2s) r 14s 7 16. - (2x 17. (a 2b 2)5 · (a -5)2 b 10 ) -8x 15 5 3 7 PRACTICE AND PROBLEM SOLVING 18–21 22 23–28 29–34 x _ Simplify. Independent Practice For See Exercises Example 18. 33 · 2 3 · 3 2 3 · 3 4 19. 6 · 6 2 · 6 3 · 6 2 6 8 20. a 5 · a 0 · a -5 1 1 2 3 4 21. x 7 · x -6 · y -3 y 3 22. Geography Rhode Island is the smallest state in the United States. Its land area is about 2.9 × 10 10 square feet. Alaska, the largest state, is about 5.5 × 10 2 times as large as Rhode Island. What is the land area of Alaska in square feet? Write your answer in scientific notation. 1.595 × 10 13 ft 2 Extra Practice _1 Simplify. Skills Practice p. S16 (2 3)3 2 9, or 512 6 26. (b 4) · b b 25 23. Application Practice p. S34 29. (3x) 3 32. 27x 24. 27. 3 30. (-4x ) 256x 12 3 4 33. (3 6)0 1 12 4 3 a b · (a 3) · (b -2) b5 (5w 8)2 25w 16 4 - (4x 3) -256x 12 (x 2)-1 2 x (x 4)2 · (x -1)-4 x 12 25. _ 28. 34. (p 4q 2) 7 p 28q 14 (x 3y 4)3 · (xy 3)-2 (a 2b )4 = a 8b 12 3 31. x 7y 6 Find the missing exponent in each expression. 35. a a 4 = a 10 6 1 38. (a 3b 6) = _ -3 a 9b 18 36. (a )4 = a 12 3 37. 39. 1 (b 2)-4 = _ 8 40. a · a 6 = a 6 0 b Geometry Write an expression for the area of each figure. 2x 3 42. 41. ÓÝ a 2b 4 43. 2m 10n 6 Ê®ÊÓ Î ÊLÊ Ê Ê ÓÊÊ{ Ê ÊÓ®Ê ÓÊ Ê>ÊÓÊLÊ Ê Ê ÊÝÊÓÊ Simplify, if possible. 44. x 6y 5 x 6y 5 47. 45. (5x 2)(5x 2) 2 125x 6 48. 7-3 Reading Strategies 7-3 READING STRATEGIES Use a Table To multiply powers with the same base, keep the base and add the exponents. A11NLS_c07_0474-0480.indd 478 Product of a Power To find the power of a power, keep the base and multiply the exponents. Power of a Power To find the power of a product, apply the exponent to each factor. Power of a Product 3 6 y6 (x 2y 2)2(x 2y)-2y 2 52. Multiplication Properties of Exponents 3 y7 x−5 • x4 • y7 = x−5 + 4 • y7 = x−1y7 = x (4 • 4 • 4) (4 • 4 • 4 • 4 • 4) 48 2 −6 (x ) = x 2 • −6 =x −12 1 = 12 x (5 • 3)2 = 52 • 32 = 25 • 9 = 225 (−4b)3 = (−4)3 • b3 = −64b3 (x0 • y−1)5 = (x0)5 • (y−1)5 = x0 • y−5 = 1 y5 In general, (ab)n = anbn a4 + (−2) • b5 43 + 5 a2 • b5 (a ) = a 88 (a ≠ 0, m and n are integers.) Simplify (x5)4 • y. x5 • 4 • y 3•2 x20y 26 multiply add Simplify. 2. 8−2 • 53 • 86 1. 23 • 24 27 8 84 • 53 4 10 7. (3v5)2 9v 10. −(4y7)2 −16y 14 7 4. m • n • m m 15 n 4 7. (5−3)3 • 40 6. 84 • 84 mn Simplify (23)2. 2 9. (2 • 9)3 5832 a2b5 m n (23)2 Simplify each expression. c (43) (45) You can use the Power of a Power Property to find a power raised to another power. Power of a Product; with both properties, a number is applied to all parts. 5 Simplify a4 • b5 • a−2. Simplify (43) (45). 48 4. 3(x + 4) = 3x + 12 shows how the Distributive Property of Multiplication is used to simplify an expression. Which property of exponents is similar to the Distributive Property? Why? d8 12/10/09 8:47:00 PM Count the number of factors. The number of factors is the exponent. am • an = am + n (a ≠ 0, m and n are integers.) In general, (am)n = am • n Power of a Product 24 Expand each factor. Or you can use the Product of Powers Property: (43)5 = 43 • 5 = 415 (89)0 = 89 • 0 = 80 = 1 What do you do with the exponents to simplify 64 • 68? 5. (m3)8 m 5 Simplify (4 ) (4 ). In general, am • an = am + n 3. What is the name of the property that can be used to simplify (6t)−9? 8. c−2 • d 8 • c−3 7-3 RETEACH 7-3 You can multiply a power by a power by expanding each factor. (43) (45) 43 • 45 = 43 + 5 = 48 38 • 56 • 3−6 = 38 + (−6) • 56 = 32 • 56 1. What do you do with the exponents to simplify (c4)−2? 2. Review for Mastery LESSON Complete each of the following. Chapter 7 0 _1 Chapter 7 Exponents and Polynomials LESSON 478 49. a · a · 3a 3a 3 1000 The table below summarizes the multiplication properties that are needed to simplify expressions with powers. KEYWORD: MA7 Resources 46. x 2 · y -3 · x -2 · y -3 3 51. 10 2 · 10 -4 · 10 5 10 , or 3 -2 50. (ab) (ab) ab 478 (2x 2) 2 · (3x 3) 3 108x 13 4 4 - (x 2) (-x 2) -x 16 4 2 5. (6 ) 3. 24 • 35 • 28 • 3−2 212 • 33 6. (4−3)2 1 68 46 8. (x2)−4 • y−3 9. (u5)−2 • (v3)4 1 1 v 12 59 x8 y3 u 10 Astronomy The graph shows the approximate time it takes light from the Sun, which travels at a speed of 1.86 × 10 5 miles per second, to reach several planets. Find the approximate distance from the Sun to each planet in the graph. Write your answers in scientific notation. (Hint: Remember d = rt.) Sunlight Travel Time to Planets Earth Planet 53. 53. Earth: 9.3 × 10 7 mi; Mars: 1.4136 × 10 8 mi; Jupiter: 4.836 × 10 8 mi; Saturn: 8.928 × 10 8 mi b. Simplify. 59. 15m 12n 9 59. 62. 2600 Jupiter 4800 Saturn 0 1000 2000 3000 4000 5000 Time (s) /////ERROR ANALYSIS///// Explain the error in each simplification below. What is the correct answer in each case? a. x 2 · x 4 = x 8 56. 760 Mars 54. Geometry The volume of a rectangular prism can be found by using the formula V = wh where , w, and h represent the length, width, and height of the prism. Find the volume of a rectangular prism whose dimensions are 3a 2, 4a 5, and 4a 2b 2. 48a 9b 2 55. 500 15 _ (-3x 2)(5x -3) - x (3m 7)(m 2n)(5m 3n 8) (2 2)2(x 5y)3 16x 15y 3 (x 4)5 = x 9 c. a _ (x 2)3 = x 2 3 = x8 7 57. (a 4b)(a 3b -6) b 5 -2 5 16 60. (b 2) (b 4) b 2 63. (-t)(-t) (-t 4) t 7 58. (6w 5)(2v 2)(w 6) 12v 2w 11 61. (3st)2t 5 9s 2t 7 64. (2m 2)(4m 4)(8n)2 Science Link For Exercise 53, you can explain to students that a planet’s distance from the Sun varies at different points in its orbit. This is because the planets have elliptical orbits. At a point called perihelion, a planet is closest to the Sun. At aphelion, it is farthest from the Sun. The root in both words, helion, comes from the Greek word for Sun. Exercise 75 involves using the formula for the speed of light. This exercise prepares students for the Multi-Step Test Prep on page 494. Answers 55a. Exponents are multiplied but should be added; x6. b. Exponents are added but should be multiplied; x20. 512m 6n 2 65. Estimation Estimate the value of each expression. Explain how you estimated. a. ⎡⎣(-3.031) 2⎤⎦ 3 b. c. Exponent is written as a power but should be multiplied; x6. (6.2085 × 10 2) × (3.819 × 10 -5) 66. Physical Science The speed of sound at sea level is about 344 meters per second. The speed of light is about 8.7 × 10 5 times faster than the speed of sound. What is the speed of light in meters per second? Write your answer in scientific notation and in standard form. 2.9928 × 10 8 m/s; 299,280,000 m/s 67. Yes; because of 3 2 67. Write About It Is (x 2) equal to (x 3) ? Explain. the Comm. Prop. of Mult., 68. Biology A newborn baby has about 26,000,000,000 cells. An adult has about 1.9 × 10 3 they are both times as many cells as a baby. About how many cells does an adult have? Write your equal to x 6. answer in scientific notation. 4.94 × 10 13 65a. 93, or 729; possible answer: Round -3.031 to -3. b. 0.024; possible answer: Round 6.2085 to 6 and round 3.819 to 4. 75c. Assoc. and Comm. Properties of Mult. Simplify. 2 69. (-4k) + k 2 17k 2 70. -3z 3 + (-3z)3 -30z 3 71. 72. (2r)2s 2 + 6(rs)2 + 1 2 73. (3a)2b 3 + 3(ab) (2b) 74. 15a 2b 3 10r s + 1 2 2 (2x 2)2 + 2(x 2)2 6x 4 (x 2)(x 2)(x 2) + 3x 2 x 6 + 3x 2 75. This problem will prepare you for the Multi-Step Test Prep on page 494. a. The speed of light v is the product of the frequency f and the wavelength w (v = fw). Wavelengths are often measured in nanometers. Nano means 10 -9, so 1 nanometer = 10 -9 meters. What is 600 nanometers in meters? Write your -7 m answer in scientific notation. 6 × 10 b. Use your answer from part a to find the speed of light in meters per second if f = 5 × 10 14 Hz. 3 × 10 8 m/s c. Explain why you can rewrite (6 × 10 -7) (5 × 10 14) as (6 × 5) (10 -7) (10 14). 7-3 PRACTICE A 7-3 PRACTICE C Practice B LESSON 7-3 Multiplication Properties of Exponents 479 7-3 7-3 PRACTICE B Multiplication Properties of Exponents Simplify. 4 LESSON 7-3 Problem Solving 7-3 PROBLEM SOLVING Multiplication Properties of Exponents Write the correct answer. 1. In the mid-nineteenth century, several landowners in Australia released domestic rabbits into the wild. Suppose 100 rabbits were released. By 1950, the population had increased about 6 106 times. Determine the wild rabbit A1NL11S_c07_0474-0480.indd 479 population in 1950. about 600,000,000 3. Saturn’s smallest moon, Tethys, has a diameter of about 6.5 102 miles. The diameter of Jupiter’s largest moon, Ganymede, is 5 times that of Tethys. Determine the diameter of Ganymede. Write your answer in standard form and in scientific notation. 2. Barnard’s star is the fifth closest star to the Earth, after the Sun and the stars in the Alpha Centauri system. It takes 1.86 108 seconds for light from Barnard’s star to reach the Earth. Light travels at a speed of 1.86 105 miles per second. Calculate the distance from Barnard’s star to the Earth. 3.46 1013 miles 4. Delaware and Montana have roughly the same population. Delaware’s area is 2.49 103 square miles. Montana is 59 times larger. Determine the area of Montana. Write your answer in standard form and in scientific notation. about 3250 mi or 147,000 sq mi or 3.25 103 mi 1.47 105 sq mi Select the best answer. 5. The formula for the volume of a cylinder is V = 2r 2 h where r is the radius and h is the height. What is the volume of the cylinder shown below? 6. What is the volume of the cube shown below? C 24x2y B 12xy2 D 36xy2 7. Belize borders Mexico and Guatemala in Central America. It has an area of 2.30 104 square kilometers. Russia borders fourteen countries and is 7.43 102 times larger than Belize. What is the area of Russia? A 1.71 106 sq km C 1.71 108 sq km B 1.71 107 sq km D 1.71 109 sq km 7-3 Challenge 1. 3 • 3 36 or 729 7-3 CHALLENGE Using Exponents to Understand Multiplication of Decimals −6 4. q • q When you learned how to multiply one decimal by another, you learned to count decimal places and move the decimal point that many places to the left. 0.003 × 0.02 Write 6 and move the decimal point 5 places to the left. 0.00006 Using properties of exponents, you can understand why this rule works. 1. 0.06 × 0.002 3. 0.15 × 0.0006 0.00012 0.00009 2. 0.04 × 0.012 4. 0.09 × 0.00012 0.00048 0.0000108 You can also find the product 0.003 × 0.02 by using a property of exponents that you learned. Notice that the final answer shown below agrees with the answer obtained by applying the rule for multiplication shown in the example above. 0.003 × 0.02 = 5. 0.06 × 0.002 0.00012 6. 0.04 × 0.012 0.00048 0.00009 8. 0.09 × 0.00012 0.0000108 G 12n9 H 64n9 J 256n9 6 0 10. (w ) 0.000006 Multiply the decimals in the same way as the whole numbers. Count the number of decimal places in each of the three numbers. Find the sum of those numbers. Move the decimal point that many places to the left. 2. 25 • 24 3. 23 • 25 • 21 29 or 512 5. r −3 4 •r •s −4 1 j4 9. (g4)−2 8. (h2)5 1 h10 2 5 11. (v ) • v f 18 16. (−5k)2 25k 2 s20 t 9 29 or 512 6. j −2 • j −4 • j 2 r s4 g8 12. (w5)−2 • w −3 4 1 v 14 1 19. (s4 t)3 • (s4 t 3) 2 Find each product by using a property of exponents. Show your work. 10. 0.04 × 0.05 × 0.003 7. c5 • b−2 • c3 1 7. 0.15 × 0.0006 0.00000036 1 q7 13. (f 6 )−4 • (f −2)−3 3 2 3 2 3×2 3×2 6 6 × = = = = 0.00006 × = 10,000 1000 100 103 102 103 × 102 103 + 2 105 9. 0.06 × 0.002 × 0.003 −1 c8 2 7/18/09 4:56:00 bPM Find each product by counting decimal places and moving the decimal point. 11. Write an extension of the rule for multiplying two decimals between 0 and 1 that applies to multiplying three such decimals. F 12n6 A 12xy LESSON 2 w 13 14. (a−2)−3 • (a5)2 a16 15. (3b)4 81b 4 18. (−3p)−2 17. −(4m)3 1 −64m 3 9p 2 20. (a2 b4)2 • (a−2 b3)−1 • a4 21. (x3 y2)−4 • (x2 y−3)−2 1 a10 b 5 x 16 y 2 22. The pitch of a sound is determined by the number of vibrations produced per second. The note “middle C” produces 2.62 × 102 vibrations per second. If a pianist plays middle C for 5 × 10−1 seconds, how many vibrations will occur? 1.31 × 102 or 131 vibrations 8. In 1989, Voyager 2 discovered six moons that orbit Neptune. The smallest of these is Naiad, which orbits Neptune in a brief 7.2 hours, or 8.22 104 years. Neptune’s orbit of the Sun takes 2 105 times longer than Naiad’s. How long does Neptune’s orbit take? F 10.2 years H 102 years G 16.4 years J 164 years Lesson 7-3 479 Students who chose B for Exercise 80 may have recalled that any nonzero number raised to the zero power is one, but they may have forgotten to multiply by x2. Critical Thinking Rewrite each expression so that it has only one exponent. (Hint: You may use parentheses.) 76. c 3d 3 In Exercise 82, 4 must be cubed because it is a factor of an expression that is being cubed. Students can eliminate choices A and B immediately because 43 is 64. (cd )3 77. 36a 2b 2 (_2ab ) 3 8a 3 78. _ b3 (6ab)2 k -2 79. _ 4m 2n 2 80. Which of the following is equivalent to x 2 · x 0? 0 1 x2 1 (_ 2kmn ) 2 x 20 5 2 81. Which of the following is equivalent to (3 × 10 ) ? 7 9 × 10 7 9 × 10 10 6 × 10 6 × 10 10 82. What is the value of n 3 when n = 4 × 10 5? 1.2 × 10 9 1.2 × 10 16 6.4 × 10 9 6.4 × 10 16 83. Which represents the area of the triangle? 6x 2 7x 2 12x 2 ÎÝ 24x 2 {Ý CHALLENGE AND EXTEND Simplify. (3 2) 3 2x x 84. 3 2 · 3 x 3 2 + x 85. 87. (x + 1)-2(x + 1)3 x + 1 88. (x + 1)2(x + 1)-3 90. (4 ) 91. (x ) x x 4x x x 2 xx 1 _ x+1 2 2 86. (x yz) x 2yz 2 3 89. (x y · x z) x 3y + 3z 92. (3x)2y 9 y x 2y Find the value of x. 93. 5 x · 5 4 = 5 8 4 94. 7 3 · 7 x = 7 12 9 3 95. (4 x) = 4 12 4 96. (6 2) x = 6 16 8 97. Multi-Step The edge of a cube measures 1.2 × 10 -2 m. What is the volume of the cube in cubic centimeters? 1.728 cm 3 Journal Have students explain how parentheses affect the expressions ab3c2, (ab3c)2 and (ab)3c2. SPIRAL REVIEW Find the value of x in each diagram. (Lesson 2-8) 98. □ABCD ∼ □WXYZ 100 99. ABC ∼ RST 15 - 7 Have students work in small groups to create a presentation that explains the properties learned in this lesson. The presentation should include visual aids as needed. ÝÊV ÎÊV Ó{ÊvÌ nÊvÌ {ÊV < ÇxÊV 9 xÊvÌ , ÝÊvÌ / Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. (Lesson 4-6) 100. 5, 1, -3, -7, … 7-3 8 101. -3, -2, 0, 3, … no yes; d = -4; -11, -15, -19 103. 7,800,000 Write each number in standard form. (Lesson 7-2) 104. 0.000495 103. 7.8 × 10 6 104. 4.95 × 10 -4 102. 0.4, 1.0, 1.6, 2.2, … yes; d = 0.6; 2.8, 3.4, 4.0 6,000,000 105. 983 × 10 -1 98.3 106. 0.06 × 10 8 Simplify. 480 Chapter 7 Exponents and Polynomials 36 2. z4 · z -2 · z z3 2 5 1 3. (x3) x6 4. -(t -3) -_ t15 3 2 9 _ 5. (5g) 125g3 6. (-3f -4) f 8 A1NL11S_c07_0474-0480.indd 3 -2 1 _ 7. (x2y) · (x3y2) y 1. 32 · 34 8. The islands of Samoa have an approximate area of 2.9 × 103 square kilometers. The area of Texas is about 2.3 × 102 times as great as that of the islands. What is the approximate area of Texas? Write your answer in scientific notation. 6.67 × 105 km2 Also available on transparency 480 Chapter 7 480 6/25/09 9:12:25 AM 7-4 7-4 Division Properties of Exponents Organizer Pacing: Traditional 1 day Block __1 day 2 Objective: Use division properties of exponents to evaluate and simplify expressions. Who uses this? Economists can use expressions with exponents to calculate national debt statistics. (See Example 3.) Objective Use division properties of exponents to evaluate and simplify expressions. Technology Lab A quotient of powers with the same base can be found by writing the powers in factored form and dividing out common factors. GI In Technology Lab Activities <D @<I Online Edition Tutorial Videos Countdown Week 15 Notice the relationship between the exponents in the original quotient and the exponent in the final answer: 5 - 3 = 2. Quotient of Powers Property WORDS The quotient of two nonzero powers with the same base equals the base raised to the difference of the exponents. EXAMPLE 1 NUMBERS ALGEBRA 67 = 67-4 = 63 _ 64 If a is a nonzero real number and m and n are a m = a m - n. integers, then _ an Warm Up Simplify. 1 1 2. 2-3 _ or _ 8 23 -3 _ 9 v6 _ 2 -1 -2 3 ) ( 3. 3 · x 4. v w x w9 3 3 y y _ 5. 38 · 3-2 36 6. _ z z3 Write in scientific notation. 1. (x2) 3 () Finding Quotients of Powers Simplify. A 3 6 = 729 Both 3 6 and 729 are considered to be simplified. C 3 _ 8 B 32 38 = 38-2 _ 32 = 3 6 = 729 x _ 5 x5 x5 = x5-5 _ x5 = x0 = 1 D a 5b 9 = _ a 5b 9 _ 4 a 4b 4 (ab) = a5-4 · b9-4 2 3 · 3 2 · 5 7 = 2 3 -1 · 3 2 - 4 · 5 7 - 5 __ 2 · 34 · 55 = 2 2 · 3 -2 · 5 2 9 (ab) 4 Simplify. 29 4 1a. _ 27 3 2 _ y 1b. _4 1 y y3 8. 0.16 × 107 1.6 × 106 7 2·3 ·5 4 5 2 = ab 5 3 × 10-2 2 ·3 ·5 __ 2 ·5 =_ 32 4 · 25 = _ 100 =_ 9 9 = a1 · b5 7. 30 × 10-3 Also available on transparency a b _ 5 x6 (xy)2 Q: Why does _ refuse to be x2y2 simplified? 2 n _ m n _ 3 A: One is the loneliest number. 3 _ 16 35 · 24 · 43 1d. _ 1c. 2 m 5 34 · 22 · 46 (m ) n 5 4 5 7- 4 Division Properties of Exponents 481 1 Introduce A11NLS_c07_0481-0487.indd 481 e x p l o r at i o n 7-4 Motivate Division Properties of Exponents x can be 1. The expression __ 2 5 Simplified x simplified by expanding the powers Expression Form 5 and dividing out common factors: 3 x __ x x 5 _________ x x x x x x x x x 3. __ x2 2 xx x a4 ___ Use this information to complete a2 the table. y7 __ 2. Describe any patterns you see in y5 the table. 3. You can use a similar method when the exponent in the denominator is greater than the exponent in the numerator. For example, 2 xx x _________ 1 __ 1 __ _____ x5 xxxxx xxx x3 x 3. Use this information to complete this table. THINK AND DISCUSS x __ 6 8/18/09 10:15:09 AM 49 Have students write out the factors in _6 . 4 4___ •4•4•4•4•4•4•4•4 Ask them how to 4•4•4•4•4•4 simplify this quotient of powers. 43 = 64 Explain that in this lesson they will learn properties for division of powers. x5 Expression 2 x __ x5 m3 ___ m7 a2 ___ a8 6 x __ x7 Simplified Form x 3 4. Explain how the results in the second table compare to those in the first table. 10 z 5. Describe how you can use your findings to simplify ___ 15 z without first expanding the powers Explorations and answers are provided in Alternate Openers: Explorations Transparencies. KEYWORD: MA7 Resources Lesson 7-4 481 EXAMPLE 2 Dividing Numbers in Scientific Notation Simplify (2 × 10 8) ÷ (8 × 10 5) and write the answer in scientific notation. Additional Examples Example 1 Simplify. 27 x4 A. _ 25, or 32 B. _ x 2 x3 2 d4e3 C. _2 d 2e (de) 3 · 43 · 55 _ 25 D. _ 32 · 44 · 53 12 2 × 10 (2 × 10 8) ÷ (8 × 10 5) = _ 5 8 8 × 10 You can “split up” a quotient of products into a product of quotients: a ×_ c a×c =_ _ b×d b d Example: 3 ×_ 3×4 =_ 4 =_ 12 _ 5×7 5 7 35 10 8 2 ×_ =_ 8 10 5 Write as a product of quotients. = 0.25 × 10 8 - 5 Simplify each quotient. = 0.25 × 10 3 Simplify the exponent. = 2.5 × 10 -1 × 10 3 Write 0.25 in scientific notation as 2.5 × 10 -1. The second two terms have the same base, so add the exponents. Simplify the exponent. = 2.5 × 10 -1 + 3 = 2.5 × 10 2 Example 2 Simplify (3 × 1010) ÷ (6 × 106) and write the answer in scientific notation. 5 × 103 2. Simplify (3.3 × 10 6) ÷ (3 × 10 8) and write the answer in scientific notation. 1.1 × 10 -2 Example 3 EXAMPLE The Colorado Department of Education spent about 4.408 × 109 dollars in fiscal year 2004—05 on public schools. There were about 7.6 × 105 students enrolled in public school. What was the average spending per student? Write your answer in standard form. $5800 3 In the year 2000, the United States public debt was about 5.6 × 10 12 dollars. The population of the United States in that year was about 2.8 × 10 8 people. What was the average debt per person? Give your answer in standard form. To find the average debt per person, divide the total debt by the number of people. 5.6 × 10 12 total debt __ =_ number of people 2.8 × 10 8 12 = 2 × 10 12 - 8 Simplify each quotient. = 2 × 10 Simplify the exponent. 4 Write in standard form. 3. In 1990, the United States public debt was about 3.2 × 10 12 dollars. The population of the United States in 1990 was about 2.5 × 10 8 people. What was the average debt per person? Write your answer in standard form. $12,800 1 • Why can the Quotient of Powers Property not be used to simplify x4 _ ? y3 • Is the smaller exponent always subtracted from the larger exponent? Explain. A power of a quotient can be found by first writing factors and then writing the numerator and denominator as powers. 2 • How do you know if your answer is in scientific notation? EX A M P L E Write as a product of quotients. The average debt per person was about $20,000. Questioning Strategies EX A M P L E 10 5.6 × _ =_ 2.8 10 8 = 20,000 INTERVENTION EX A M P L E Economics Application 3 • How do you know which number is the divisor? Notice that the exponents in the final answer are the same as the exponent in the original expression. 482 Chapter 7 Exponents and Polynomials 2 Teach • What mathematical operations are needed to answer the question? A11NLS_c07_0481-0487.indd 482 How do you know? Guided 8/18/09 10:15:51 AM Instruction Review the multiplication properties of exponents because some problems in this lesson use both multiplication and division properties. Introduce the division properties in this lesson by examining the powers written in factored form first. Through Graphic Organizers Have students create a graphic organizer to show evidence that each property works. 279,936 67 Quotient _ =_ 1296 of Powers 64 216 Property Power of a Quotient Property 482 Chapter 7 (_35 ) 4 = (0.6) 0.1296 4 67-4 = 63 216 34 _ 81 _ = 54 625 0.1296 "" Positive Power of a Quotient Property WORDS NUMBERS A quotient raised to a positive power equals the quotient of each base raised to that power. EXAMPLE 4 () 3 _ 5 4 ALGEBRA If a and b are 4 nonzero real 3 3 3 3 3 3 · 3 · 3 · 3 _ _ _ _ _ __ = · · · = = 4 numbers and n is 5 5 5 5 5·5·5·5 5 a positive integer, () a then __ b n Ê,,", ,/ an = ___ . bn When using the Power of a Product Property in Example 4B, students might forget to apply the power to the coefficients of terms. Remind students to apply the power to each factor in the product, including variables and coefficients. Finding Positive Powers of Quotients Simplify. A (_34 ) 3 (_34 ) = _ 4 Additional Examples 3 3 Example 4 3 Use the Power of a Quotient Property. 3 27 =_ 64 B (_) ( ) 2x 3 yz 2x 3 _ yz Simplify. () Simplify. 4 A. _ 7 ( ) 3 3 (2x 3) =_ (yz)3 3 3 Simplify. 6 4 ( ) 3d2 B. _ ef 16 _ 49 2 4 81d 8 _ e4f 4 4 _ y6 Use the Power of a Product Property: (2x 3) 3 = 2 3(x 3)3 and (yz) 3 = y 3z 3. Simplify 2 3 and use the Power of a Power 3 Property: (x 3) = x 3 · 3 = x 9. 64 2 , or _ _ ( )3 2 23 4a. _ 32 2x3 C. _3 (xy) Use the Power of a Quotient Property. 2 3(x 3) =_ y 3z 3 8x 9 =_ y 3z 3 2 ( ) ab 4 4b. _ 81 c 2d 3 5 a b _ 5 20 c 10d 15 ( ) a 3b 4c. _ a 2b 2 3 INTERVENTION Questioning Strategies EX AM P LE a _ 3 4 • What happened to a variable if it was in the original expression, but not in the simplified expression? b3 1 . What if x is a fraction? Remember that x -n = _ xn (_ab ) -n () a 1 _ =_ __a n = 1 ÷ b (b) n an =1÷_ bn Write the fraction as division. Use the Power of a Quotient Property. n b =1·_ an Multiply by the reciprocal. n b =_ an Simplify. () a b Therefore, (_ = (_ a) . b) b = _ a -n n Use the Power of a Quotient Property. n 7- 4 Division Properties of Exponents A1NL11S_c07_0481-0487.indd 483 Auditory The names of the properties in this lesson sound a lot alike. Help students differentiate between the two: □ Say “Quotient,” and then write _ . □ x3 Say “of Powers,” and then write _2 . x 3 Say “Power,” and then write ( ) . 483 6/25/09 9:10:42 AM Say “of a Quotient,” and then x 3 write _ y . () Lesson 7-4 483 Negative Power of a Quotient Property WORDS Additional Examples A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power. Example 5 Simplify. () 3 A. _ 4 -3 ( ) 2x2 B. _ y3 -2 64 _ 27 y6 _ 4x4 () ( ) 2 C. _ 3 -2 6m _ 2n -3 NUMBERS EXAMPLE 5 () 2 _ 3 A (_25 ) (_25 ) = (_52 ) Questioning Strategies 5 B • How can you make a negative exponent positive? (__ba ) -n () = _ba_ n bn = ___ an . Rewrite with a positive exponent. -3 ? Use the Power of a Quotient Property. 5 3 = 125 and 2 3 = 8. ( ) ( ) ( ) 3x _ y2 3x _ y2 -3 -3 3 y2 = _ 3x Rewrite with a positive exponent. (y 2) = _3 (3x) y6 =_ 3 3x 3 y6 =_ 27x 3 3 Multiple Representations Discuss with students the advantages or disadvantages of different methods of simplifying. For example, in Example 5A, students could use the Power of a Quotient Property first: -3 If a and b are nonzero real numbers and n is a positive integer, then 3 5 =_ 23 125 =_ 8 INTERVENTION (__25 ) 34 =_ 24 -3 3 () 4 Simplify. n3 _ 12m3 1 • What is the value of __ 5 () 3 = _ 2 Finding Negative Powers of Quotients -3 EX A M P L E -4 ALGEBRA -3 53 125 2 =____ = __ = ___ . -3 8 23 C 5 Use the Power of a Quotient Property. Use the Power of a Power Property: (y 2)3 = y 2·3 = y 6. Use the Power of a Product Property: (3x) 3 = 3 3x 3. Simplify the denominator. 2x (_34 ) (_ 3y ) 3y 2x 4 _ = (_ (_34 ) (_ 3y ) 3 ) ( 2x ) -1 -2 -1 -2 1 2 Rewrite each fraction with a positive exponent. (3y)2 4 ·_ =_ 3 (2x) 2 Whenever all of the factors in the numerator or the denominator divide out, replace them with 1. Use the Power of a Quotient Property. 3 2y 2 4 ·_ =_ 3 2 2x 2 1 9 3y 2 4 ·_ =_ 2 13 14x Use the Power of a Product Property: (3y)2 = 3 2y 2 and (2x) 2 = 2 2x 2. Divide out common factors. 3y 2 =_ x2 Simplify. ( ) 4 5a. _ 32 484 -3 9 729 _ , or _ 3 43 b c (b c ) _ 16a 2a 64 5b. _ 2 3 -4 8 12 4 9s _t ( ) (_ t ) s s 5c. _ 3 -2 2 -1 4 Chapter 7 Exponents and Polynomials 3 Close A1NL11S_c07_0481-0487.indd 484 Summarize and INTERVENTION Have students give an example of the Quotient of Powers Property and the Power of a Quotient Property. Possible answers: x5 Quotient of Powers: _3 = x5-3 = x2 x x 3 _ x3 Power of a Quotient : _ y = y3 () 484 Chapter 7 Diagnose Before the Lesson 7-4 Warm Up, TE p. 481 Monitor During the Lesson Check It Out! Exercises, SE pp. 481–484 Questioning Strategies, TE pp. 482–484 Assess After the Lesson 7-4 Lesson Quiz, TE p. 487 Alternative Assessment, TE p. 487 7/18/09 4:54:37 PM Answers to Think and Discuss THINK AND DISCUSS 1. Possible answer: Both the Quotient of Powers Property and the Product of Powers Property require that the bases be the same. For quotients, you subtract the exponents. For products, you add the exponents. In the Power of a Quotient Property, each term is raised to the same power. In the Power of a Product Property, each factor is raised to the same power. 1. Compare the Quotient of Powers Property and the Product of Powers Property. Then compare the Power of a Quotient Property and the Power of a Product Property. 2. GET ORGANIZED Copy and complete the graphic organizer. In each cell, supply the missing information. Then give an example for each property. vÊ>Ê>`ÊLÊ>ÀiÊâiÀÊÀi>ÊÕLiÀÃÊ >`ÊÊ>`ÊÊ>ÀiÊÌi}iÀÃ]ÊÌ i°°° >Ê ÚÚ ÚÊÊ L > ÚÚ > >Ê ÚÊÊ L ÚÚ 2. See p. A7. 7-4 Exercises 7-4 Exercises KEYWORD: MA7 7-4 KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 p. 481 SEE EXAMPLE p. 482 SEE EXAMPLE _ 22 · 34 · 44 2 2. _ 29 · 35 3 5 8 25 1. _ 56 2 p. 482 (2.8 × 10 11) ÷ (4 × 10 8) 7 × 10 2 6. p. 483 SEE EXAMPLE p. 484 4 () _ 25 3 16 _ 13. (_ 4) 9. 2 _ 5 2 -2 5 Assign Guided Practice exercises as necessary. (5.5 × 10 3) ÷ (5 × 10 8) 7. 1.1 × 10 -5 (1.9 × 10 4) ÷ (1.9 × 10 4) 1 8. Sports A star baseball player earns an annual salary of $8.1 × 10 6. There are 162 games in a baseball season. How much does this player earn per game? Write your answer in standard form. $50,000 Simplify. SEE EXAMPLE 4 _ 2 a 5b 6 a 4. _ 3 7 a b b 15x 6 3 3. _ 5x 6 Simplify each quotient and write the answer in scientific notation. 5. 3 Assignment Guide Simplify. 9 10. 14. ( ) x2 _ xy 3 ( ) 2x _ y3 3 -4 x _ 3 y 11. 9 _ y 12 16x 15. 4 (( ) ) a3 _ 2 a 3b 2 1 _ 6 4 a b If you finished Examples 1–5 Basic 17–45, 47–57, 62–71 Average 17–49, 54–60, 62–71 Advanced 17–49, 54–71 y 10 y9 12. _ y y x 2b 16. _ _ ()( ) _ ) ( y 3a x 2 _ 3 -1 3a _ 2b -2 2 3 -4 2 2 If you finished Examples 1–3 Basic 17–25, 34–37, 47 Average 17–25, 34–37, 47, 59 Advanced 17–25, 34–37, 47, 59 8 Homework Quick Check 12 Quickly check key concepts. Exercises: 18, 22, 25, 28, 30, 48 PRACTICE AND PROBLEM SOLVING Simplify. 3 9 27 17. _ 36 5 4 · 3 3 75 18. _ 52 · 32 x 8y 3 x5 19. _ x 3y 3 _ 3 x 8y 4 y 20. _ 9 xz x yz Simplify each quotient and write the answer in scientific notation. 21. 23. Independent Practice (4.7 × 10 -3) ÷ (9.4 × 10 3) 5 × 10 -7 (4.2 × 10 -5) ÷ (6 × 10 -3) 7 × 10 -3 22. 24. (8.4 × 10 9) ÷ (4 × 10 -5) 2.1 × 10 14 (2.1 × 10 2) ÷ (8.4 × 10 5) 2.5 × 10 -4 7- 4 Division Properties of Exponents A1NL11S_c07_0481-0487.indd 485 485 7/18/09 4:54:43 PM KEYWORD: MA7 Resources Lesson 7-4 485 Social Studies Link For Exercise 47, students might be interested to know that the population density of the United States is actually rather low. Some cities are dense, but there is a lot of open land as well. In comparison, Puerto Rico has a population density of 435 people/km2. Bermuda’s population density is even higher, at 1211 people/km2. Independent Practice For See Exercises Example 17–20 21–24 25 26–29 30–33 1 2 3 4 5 25. Astronomy The mass of Earth is about 3 × 10 -3 times the mass of Jupiter. The mass of Earth is about 6 × 10 24 kg. What is the mass of Jupiter? Give your answer in scientific notation. 2 × 10 27 kg Simplify. 26. 16 (_23 ) _ 81 27. ( ) 30. (_17 ) 31. ( ) 4 Extra Practice Skills Practice p. S16 Application Practice p. S34 -3 343 3 a4 _ b2 x2 _ y5 a _ 28. ( ) 32. 2 _ 8w (_ 16 ) w 12 b6 y _ 25 -5 x 10 a _ 29. ( ) 33. 196 _ 6x (_14 ) (_ 7) 9x 12 6 a 3b 2 _ ab 3 b6 7 -1 7 y _ 3 xy 2 _ x 3y 3 x6 -2 -2 2 Simplify, if possible. Answers 47. 2000: 3 × 101 1990: 2.65 × 101 49. Possible answer: The bases are the same, so subtract the exponent of the denominator from the exponent of the numerator: 3 (5x 2) 25x 4 38. _ 5x 2 27 ( ) a ( ) _ a 10 1 44. _ 43. (_ )_ ( x y ) _1 42. ( ) b -2 _ b3 2 c 2a 3 _ a5 39. 1 _ 2 c _ 4 4 2 10 -5 · 10 5 b 10 2 2 -3 -1 x 2y 100 ( ) 41. 6 -2 -p 4 _ -5p 3 3 3a _ a3 · a0 40. _ (3x 3) 3x 5 37. _2 (6x 2) 4 8d 5 2d 2 35. _ 4d 3 3 1995: 2.84 × 101 _ 2 3 x 2y 3 x y 36. _ a 2b 3 a 2b 3 x6 x 34. _ x5 25 _ p2 (-x 2) 45. _4 -1 - (x 2) 4 y3 46. Critical Thinking How can you use the Quotient of a Power Property to explain 1 = x 0 = x 0 - n = x -n x0 1 the definition of x -n ? Hint: Think of __ as __ . xn xn ( 45 __ = 43 = 64. 2 4 _ _ ) xn 47. Geography Population density is the number of people per unit of area. The area of the United States is approximately 9.37 × 10 6 square kilometers. The table shows population data from the U. S. Census Bureau. 42 When simplifying __ , subtracting 45 the exponents gives a negative 42 1 1 exponent: __ = 4-3 = __ = ___ . 64 45 43 xn United States Population Year Population (to nearest million) 2000 2.81 × 10 8 1995 2.66 × 10 8 Write the approximate population density 1990 2.48 × 10 8 (people per square kilometer) for each of the given years in scientific notation. Round decimals to the nearest hundredth. 48. Chemistry The pH of a solution is a number that describes the concentration of hydrogen ions in that solution. For example, if the concentration of hydrogen ions in a solution is 10 -4, that solution has a pH of 4. Lemon juice pH 2 Apples pH 3 Water pH 7 Ammonia pH 11 a. What is the concentration of hydrogen ions in lemon juice? 10 -2 b. What is the concentration of hydrogen ions in water? 10 -7 c. How many times more concentrated are the hydrogen ions in lemon juice than in water? 10 5, or 100,000, times more concentrated 5 2 4 4 49. Write About It Explain how to simplify __ . How is it different from simplifying __ ? 5 2 4 4 Find the missing exponent(s). 7-4 PRACTICE A 7-4 PRACTICE C ________________________________________ ___________________ __________________ 7-4 PRACTICE B 7-4 486 Division Properties of Exponents Simplify. 7 6 = 67 – 5 = 6 2 = 36 65 2. 9 3. 4. 7 ( ) a2 _ b 4 8 a 3 =_ b 12 ( ) x4 _ y 53. j j8 5. § s3 t · 8. ¨ 4 ¸ © st ¹ s 1 3 x –3 7 t § 3a · 10. ¨ ¸ © 2b ¹ –4 27 8 − 81a 4 –2 49t § 4s · < ¨ ¸ © 6t ¹ –2 § 3c · 13. ¨ ¸ © –2 ¹ 2 − 16s 2 –1 §d · ¨4¸ © ¹ 15. (3.8 × 10 ) ÷ (1.9 × 10 ) 2 × 1011 3 81v 4 t4 300,000 yards 18. It takes 5 yards of fabric to manufacture a dress. If the textile factory turned their entire yearly production of 1.08 × 108 yards of fabric into dresses, how many could they make? Give your answer in scientific notation. Chapter 7 –4 16. (2.5 × 10 ) ÷ (5 × 10 ) 17. A textile factory produces 1.08 × 108 yards of fabric every year. If the factory is in operation 360 days a year, what is the average number of yards of fabric produced each day? Give your answer in standard form. 486 45 = 45 – 3 = 42 = 16 43 7-4 RETEACH 7-4 In general, am = am – an x12 y 4 x11 = x12 – 1 . y 4 – 6 = x 11 y – 2 = 2 xy 6 y Review for Mastery LESSON n Division Properties of Exponents The Quotient of Powers Property can be used to divide terms with exponents. am = am – n (a 0, m and n are integers.) an 75 x 7y Simplify 2 . Simplify . 7 x3 75 x7y 5–2 =7 = x7 – 3 • y 72 x3 = 73 To find the negative power of a quotient, apply the positive exponent to the numerator and denominator of the reciprocal. Negative Power of a Quotient –4 4 In general, n an a = n b b In general, 2 3 –5 5 35 243 3 = = 5 = 32 2 2 x3 y 5 x –4 4 x5 x 20 x8 = 3 = 12 4 = 4 x y y x y a b –n b a n = bn = n a = x4y The Positive Power of a Quotient Property can be used to raise quotients to positive powers. n an a = n (a 0, b 0, n is a positive integer.) b b 3 4 2x 5 Simplify 4 . y 2 Simplify . 5 3 4 24 2 = 4 5 5 16 = 625 Use the Positive Power of a Quotient Property. b13 1. What do you do with the exponents to simplify 8 ? subtract b 2. 5 Rewrite the expression 8 –4 using a positive exponent. 8 5 1. 56 54 2. 3 Positive Power of a Quotient 4. 12 128 3 10. 2 2 5. 5 144 f 3g 4 7. 7 5 f gh –3 g3 4 f h 8 27 5 st 8. 4 t 16 625 x y 11. 3 xy 3 t 18 6 s 2 6 x4 y4 9 6. 8 64 81 –2 2c 2 9. 4 c d 5f 12. 7 g 3 5 –2 23 53 32 10 c d 5 g 14 25f 6 x6y 5 y3 23 ( x 5 )3 ( y 4 )3 = 8x15 y 12 Use the Power of a Product Property. Simplify. 3. x6 y2 25 2 4. 5 Simplify each expression. 4 = Use the Positive Power of a Quotient Property. Simplify. 4 4t 3. What is the name of the property that can be used to simplify the expression ? 5 10 2x 5 (2 x 5 )3 4 = ( y 4 )3 y Simplify. Complete each of the following. 5 × 106 2.16 × 107 dresses –4 y3 = _ 3; 4 x Name _______________________________________ Date___________________ Class __________________ 5 81m n 16 32 To divide powers with the same base, keep the base and subtract the exponents. To find the 32 25 2 positive power of a = 5 = Positive 3 243 3 quotient, apply the power exponent to the x 2 y 5 3 ( x 2 y 5 )3 x 6 y 15 of a Quotient = = x6y12 y = numerator ( y )3 y3 and denominator. 6 4 3cd 2 –6 A1NL11S_c07_0481-0487.indd 486 4 Simplify. Write the answer in scientific notation. 5 Quotient of Powers 2 § § 3mn · –1 · 14. ¨ ¨ ¸ ¨ © 2 ¸¹ ¸ © ¹ –2 __________________ 7-4 READING STRATEGIES Use a Table The table below summarizes the multiplication properties that are needed to simplify expressions with powers. § –t · 11. − ¨ ¸ © 3v ¹ 16b 4 ___________________ Reading Strategies 7-4 5m 3 ( x 4 )2 7. ( x 3 )5 c _______________________________________ LESSON 20m 4m 2 -1 Chapter 7 Exponents and Polynomials 5 =t 5 j6 c3 d 2 6. 2 5 c d §6· 12. ¨ ¸ ©7¹ Š 1 w7 d t 12 = t 12 t7 2 w w2 §2· 9. ¨ ¸ ©3¹ 52. Practice B LESSON 1. x7 = x4 3 51. _ x x = x2 6 50. _ x4 3 x3 5. 2 y or 8 125 a 7. 2 b 3 6 x 18 x 8. xy 3m 3 6. 2 n 9m 6 y 12 3 a2b 4 3 (ab ) b a n4 2 30 9. 20 2 2 7/18/09 4:54:57 PM 54. This problem will prepare you for the Multi-Step Test Prep on page 494. a. Yellow light has a wavelength of 589 nm. A nanometer (nm) is 10 -9 m. What is 589 nm in meters? Write your answer in scientific notation. 5.89 × 10 -7 m b. The speed of light in air, v, is 3 × 10 8 m/s, and v = fw, where f represents the frequency in hertz (Hz) and w represents the wavelength in meters. What is the frequency of yellow light? about 5.09 × 10 14 Hz 55. Which of the following is equivalent to (8 × 10 6) ÷ (4 × 10 2)? 2 × 10 3 2 × 10 4 4 × 10 3 ( ) 12 x 56. Which of the following is equivalent to _ 3xy 4 9y 8 _ x 22 Students who chose C or D for Exercise 55 may have subtracted 4 from 8 instead of dividing. 4 × 10 4 For Exercise 56, remind students that the exponent outside the parentheses applies to every factor inside them, including 3. -2 3y 8 _ x 22 ? 3y 6 _ x 12 6y 8 _ x 26 In Exercise 57, point out that the negative sign in the denominator will be unaffected by the exponent because it is outside the parentheses. (-3x) _ ? 4 57. Which of the following is equivalent to -1 - (3x)4 1 _ 81x 4 -81x 4 1 Exercise 54 involves dividing numbers written in scientific notation. This exercise prepares students for the Multi-Step Test Prep on page 494. CHALLENGE AND EXTEND 58. Geometry The volume of the prism at right is V = 30x 4y 3. Write and simplify an expression for the prism’s height in terms of x and y. 2x 2y 2 xÝÞ ÎÝ _ (x + 1) 2 1 60. Simplify _3 . (x + 1) x + 1 3 2x . 59. Simplify _ 3 2x -1 3 61. Copy and complete the table below to show how the Quotient of Powers Property can be found by using the Product of Powers Property. Statements = am · a-n 2. 1 =a ·_ an m a = _ m 3. 4. Have students compare and contrast multiplying and dividing two numbers that are each written in scientific notation. Reasons + 1. a m-n = a Journal an m; -n 1. Subtraction is addition of the opposite. 2. Product of Powers Property Have students simplify 3 ( ) -3 ? Def. of neg. exp. 3. −−−−−−−−−−−−−−−−−−−−−−− (5x2y) 4. Multiplication can be written as division. step which property or definition they used. 2xy · ___ z4 and explain at each SPIRAL REVIEW Find each square root. (Lesson 1-5) 6 63. √ 1 1 62. √36 64. - √ 49 -7 Solve each equation. (Lesson 2-4) _ 7-4 65. √ 144 12 Simplify. _ 66. -2(x -1) + 4x = 5x + 3 - 1 67. x - 1 - (4x + 3) = 5x - 1 3 2 Simplify. (Lesson 7-3) 68. 3 2 · 3 3 3 5, or 243 69. k 5 · k -2 · k -3 1 70. (4t 5) 2 16t 10 71. -(5x 4) -125x 12 3 7- 4 Division Properties of Exponents _______________________________________ LESSON 7-4 ___________________ __________________ Problem Solving 7-4 PROBLEM SOLVING LESSON Division Properties of Exponents 7-4 Write the correct answer. 1. Kudzu is a fast-growing vine that has become a nuisance in the southeastern United States. It covers 2.5 105 acres in Alabama. In 2004 the population of Alabama was estimated to be 6 A1NL11S_c07_0481-0487.indd 4.45 487 10 people. How many acres of kudzu are there for each person in Alabama? 0.056 acres 3. Voyager 2 was launched in 1979 to explore the planets of the outer solar system. The spacecraft travels an average of 4.68 106 kilometers in one year. Determine the speed of Voyager 2 in kilometers per hour. (Hint: 1 year = 8760 hours) 5.34 102 km/h Name _______________________________________ Date___________________ Class __________________ 2. A cylindrical water tank has a volume of 6x2y4 cubic meters. The formula for the volume of a cylinder is r 2h. The water tank has a radius of xy meters. What is its height? 6y2 meters 4. The population of Laos is 6.22 106. In 2004 its gross domestic product (GDP) was $1.13 1010. The population of Norway is 4.59 106. In 2004 its GDP was $1.83 1011. What is the GDP per capita, or per person, of Laos and Norway? Laos: $1817; Norway: $39,869 6. A storage chest is shaped like a cube. What is the volume of the storage chest? 7-4 CHALLENGE Applying Properties of Exponents to Rational Numbers You can use the following three facts to discover a new and interesting fact about rational numbers: • A rational number is the quotient of two integers with a nonzero denominator. • Every integer can be written as a product of powers of prime numbers, called the prime factorization of the given number. For example, 120 = 233151. • When dividing two powers with the same base, subtract the exponents. 1 105 103 = 102 and = 103 105 102 Write the prime factorization of each integer. 23 31 1. 24 3. 452 22 1131 5. 18 24 6. 48 180 2 • 32 = 3 22 • 3 22 24 • 3 = 22 22 • 32 • 5 3 • 5 7. 250 288 2 • 53 = 53 25 • 32 24 • 32 8. 540 1800 22 • 33 • 5 = 3 23 • 32 • 52 2 • 5 2. 108 4. 1800 22 33 23 32 52 9. Examine the final quotients that you wrote in Exercises 5–8. Explain why a prime-number base that appears in a numerator does not appear in the denominator and why a primenumber base that appears in a denominator does not appear in the numerator. A 5b2 yards C B 5b3 yards D 25b6 yards 5b6 yards 7. The wavelengths of electromagnetic radiation vary greatly. Green light has a wavelength of about 5.1 107 meters. The wavelength of a U-band radio wave is 2.0 102 meters. About how much greater is the wavelength of a U-band radio wave than that of green light? A 2.55 109 C B 2.55 105 D 3.92 105 3.92 104 F x3 cubic units 64 H 32 cubic units x3 x3 cubic units J 64x 3 cubic units G 32 8. Puerto Rico has an area of 5.32 103 square miles and a population of 3.89 106. What is the population density of Puerto Rico in persons per square mile? 3 F 1.37 10 G 1.37 102 H 2 7.31 10 J 7.31 103 ( ) If a prime number base b appear in the numerator (or denominator), it cannot occur in the denominator (or numerator) as well because then the rational number bna bn m a is not fully simplified. ex: m = c b c a a 10. Let b be a rational number. Write a generalization about the representation of b as the quotient of prime numbers raised to powers. Illustrate your generalization by using 21 33 54 120 = 23 31 51 . Every rational number can be written as a quotient whose numerator is 1 or the product of prime numbers raised to positive integer exponents and whose denominator can be written as 1 or the product of prime numbers raised to positive integer exponents, and there are no prime bases common to the numerator and the denominator. 8y3 _ x6 x5y2 2. _3 (xy) x2 _ y n ( ) _ 16m 8c 3 _ 9d _ 5. (_ c ) ( 2c ) 81d 4m 4. _ n3 2 -2 6 2 2 -3 6 ) ÷ (5 × 6. Simplify (3 × 105) and write the answer in scientific notation. 6 × 106 6/25/09 9:10:58 AM For each rational number, write the numerator and denominator by using the prime factorization of each. Then use the Quotient-of-Powers Property to simplify the result. Do not multiply out the powers of prime numbers that remain. Select the best answer. 5. A rectangular parking lot has an area of 10a3b6 square yards. What is the width of the parking lot? Challenge 487 48 1. _3 45 4 3 2xy2 3. _ x3y 1012 7. The Republic of Botswana has an area of 6 × 105 square kilometers. Its population is about 1.62 × 106. What is the population density of Botswana? Write your answer in standard form. 2.7 people/km2 Also available on transparency Lesson 7-4 487 7-5 Organizer 7-5 Pacing: Traditional 1 day Block Rational Exponents __1 day 2 Objective: Evaluate and simplify GI expressions containing rational exponents. <D @<I Why learn this? You can use rational exponents to find the number of Calories animals need to consume each day to maintain health. (See Example 3.) Objective Evaluate and simplify expressions containing rational exponents. Online Edition Vocabulary index Tutorial Videos, Interactivity Recall that the radical symbol √ is used to indicate roots. The index is the small number to the left of the radical symbol that tells which root to take. 3 3 = 2. For example, √ represents a cube root. Since 2 3 = 2 · 2 · 2 = 8, √8 Countdown Week 16 Another way to write nth roots is by using exponents that are fractions. = b k. For example, for b > 1, suppose √b √ b = bk ( √b )2 Warm Up 5 2. √0 6 3. √ 64 4 4. √1 n When b = 0, √ b = 0. n b = 1. When b = 1, √ 0 3 4 So for all b > 1, √ b= Power of a Power Property 1 = 2k If b m = b n, then m = n. 1 =k _ 2 Divide both sides by 2. 1 _ b 2. _1 1 5 5. √100,000 2 b 1 = b 2k Simplify each expression. 36 1. √ = (b k) Square both sides. Definition of b n WORDS 10 NUMBERS A number raised to the 1 power of __ n is equal to the nth root of that number. 3 6. - √ 27 -3 Also available on transparency _1 3 2 = √3 _1 4 5 4 = √ 5 _1 27 EXAMPLE and this textbook Q: What do 3 √7 have in common? 1 = √ 2 7 ALGEBRA If b > 1 and n is an integer, where n ≥ 2, _1 n then b n = √ b. _1 _1 3 b3 = √ b, b 2 = √b, _1 4 and so on. b 4 = √b, _1 Simplifying b n Simplify each expression. _1 A 125 3 A: Both have an index. _1 _1 3 3 = √ 125 3 = √125 53 =5 √ is equivalent 2 to √. See Lesson 1-5. _1 _1 _1 _1 Use the definition of b n . B 64 6 + 25 2 _1 6 64 6 + 25 2 = √ 64 + √ 25 Use the definition of b n . 6 6 = √2 + √ 52 = 2 + 5 =7 488 Chapter 7 Exponents and Polynomials 1 Introduce e x p l o r at A1NL11S_c07_0488_0493.indd 488i o n Motivate 7-5 Rational Exponents You will need a graphing calculator for this Exploration. Recall that whole-number exponents mean repeated multiplication. For example, 3 4 3 · 3 · 3 · 3 81. You use a radical 4 to show the inverse operation: 兹81 3. 1. You can use your calculator to evalu3 ate radicals. To find 兹4096 , enter 3 and x then press . Select 5: 兹 , enter 4096, and press . 2. Use your calculator to help you complete the table. 3 You can also use your calculator to explore fractional exponents. To evaluate 1 __ 4096 3, first enter 4096. Then press 1 3 and press . 4 Use your calculator to help you complete the table. %.4%2 Radical Value 兹 4096 3 兹4096 4 兹4096 6 兹4096 Power 1 __ 1 __ 4096 4 1 __ 4096 6 THINK AND DISCUSS Chapter 7 Value 4096 2 4096 3 5. Describe what you notice about the two tables. 1 1 __ Then write 4 2 2 on the board. Ask students to guess the value of this expression and explain their thinking. Possible answer: 32; the value should be between 16 and 64. Tell students that in this lesson they will learn why 1 __ 4 2 2 = 32. 1 __ KEYWORD: MA7 Resources 488 16 Write 42 and 43 on the board and have students simplify both expressions. 16 and 64 16 Explorations and answers are found in Alternate Openers: Explorations Transparencies. 7/18/09 4:53:25 PM "" Simplify each expression. 1 _ 1a. 81 4 1 _ 1 _ 3 1b. 121 2 + 256 4 15 2 _ A fractional exponent can have a numerator other than 1, as in the expression b 3. You can write the exponent as a product in two different ways. 2 _ 1 ·2 _ 2 _ b3 = b3 = b3 = b ( ) 1 2 _ b3 = ( √ b) 3 Power of a Power Property _1 2 1 2·_ 3 With fractional exponents with a numerator other than 1, as in Example 2, students may confuse the index with the power. Write power _ base index on the board for students to use as a reference. 1 _ b2 3 =( Ê,,", ,/ ) 2 = √b 3 Definition of b n m _ Definition of b n Additional Examples WORDS NUMBERS ALGEBRA 3 ) = 2 2 = 4 8 3 = ( √8 If b > 1 and m and n are integers, where m ≥ 1 and n ≥ 2, then _2 A number raised to the m power of __ n is equal to the nth root of the number raised to the mth power. 2 _2 3 3 2 = √ 8 3 = √8 64 = 4 b m _ n = ( √ b) = m n Example 1 Simplify each expression. 1 _ m √ n A. 343 3 b . 7 1 _ 5 1 _ B. 32 + 9 2 2 EXAMPLE Simplifying Expressions with Fractional Exponents Simplify each expression. _2 Example 2 Simplify each expression. _4 B 32 5 A 216 3 5 _ 5 4 A. 81 4 3 3 = √6 = ( √ 25) 4 B. 3125 = (6)2 = 36 = (2)4 = 16 _2 m _ 216 3 = ( √ 216 ) 2 3 ( Definition of b n )2 _4 32 5 = ( √ 32 ) 5 2a. 3 EXAMPLE 8 2b. 2 _ 15 1 2c. 4 _ 27 3 243 2 _ 5 25 Example 3 Given a cube with surface area S, the volume V of the cube can be found by using the formula S _3 V = _ 2 . Find the volume of a 6 cube with surface area 54 m2. 27 m3 Simplify each expression. 3 _ 16 4 5 81 ( ) Biology Application The approximate number of Calories C that an animal needs each day is _3 given by C = 72m 4 , where m is the animal’s mass in kilograms. Find the number of Calories that a 16 kg dog needs each day. _3 C = 72m 4 _3 = 72(16) 4 Substitute 16 for m. = 72 · ( √ 16 ) 4 3 m _ INTERVENTION Definition of b n Questioning Strategies = 72 · ( √ 24) 4 3 = 72 · (2) 3 EX AM P LE • How do you change a fractional exponent to an nth root? = 72 · 8 = 576 The dog needs 576 Calories per day to maintain health. 3. Find the number of Calories that an 81 kg panda needs each day. 1944 7- 5 Rational Exponents 489 EX AM P LE 2 • Will you get the same answer if you raise the number to the power before taking the root? EX AM P LE 2 Teach 3 • How can you use the order of operations to solve this problem? A1NL11S_c07_0488_0493.indd 489 7/20/09 4:50:57 PM Guided Instruction Review powers and roots by writing 3 53 = 5 • 5 • 5 = 125 and √ 125 = 5. Have students practice writing several of these examples. Then present the definition of 1 _ bn 1 and discuss Example 1. Show students 1 _ 1 _ two special cases: 1 n = 1 and 0 n = 0 for all natural-number values of n. Continue m _ with the definition of b n and the remaining examples. Remind students of the properties of exponents before presenting Example 4. Through Cooperative Learning Have students work in pairs. Students take turns rolling both a red (r) and a blue (b) number cube. After each roll, the student uses the numbers shown on the cubes to r _ complete the expression 64 b . Then the student simplifies the expression or states that it cannot be simplified. The other student checks the answer and then rolls the number cubes to decide the next expression. Multiple Representations In Example 2, the power can also be placed under the radical 2 _ sign: 216 3 = √ 216 2 . However, it is usually more convenient to evaluate the root and then evaluate the power. 3 Lesson 7-5 489 Remember that √ always indicates a nonnegative square root. When you simplify variable expressions that contain √, such as √ x 2 , the answer cannot be negative. But x may be negative. Therefore you simplify √ x 2 as ⎪x⎥ to ensure the answer is nonnegative. Additional Examples Example 4 Simplify. All variables represent nonnegative numbers. 4 4b20 ab5 A. √a 1 _ B. (x6y 4) 2 √ y2 When x is... and n is... x n is... n n is... and √x Positive Even Positive Positive Negative Even Positive Positive x3y3 Positive Odd Positive Positive Negative Odd Negative Negative n to ⎪x⎥, because you do not know When n is even, you must simplify √x n whether x is positive or negative. When n is odd, simplify √ x n to x. n INTERVENTION Questioning Strategies EXAMPLE Using Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. 4 EXA M PL E 4 A • How do you use properties of exponents to simplify expressions? x 9y 3 √ _1 3 x 9y 3 = (x 9y 3) 3 √ 3 _1 Definition of b n _1 _1 = (x 9) 3 · (y 3) 3 • How can you check your answer after you have simplified an expression? When you are told that all variables represent nonnegative numbers, you do not need to use absolute values in your answers. = = (x B Power of a Product Property ( )·( ) _ _ 9· 1 x 3 3· 1 y 3 ) · (y ) = x 3 1 Power of a Power Property 3 y Simplify exponents. ( _) √y (x y ) √y = (x y_) · y _ = (x ) · (y ) · y 1 4 3 x 2y 2 1 4 _ 3 2 2 3 1 4 2 2 3 Power of a Product Property ) · (y ) · y 8 =x ·y 8 3 1 ·4 2 2·4 = (x √y3 = y 2 2+1 Simplify exponents. =x y 8 3 Product of Powers Property Simplify. All variables represent nonnegative numbers. 4a. Answers to Think and Discuss 5 _ 1 2 _ 2 4 x 4y 12 xy 3 √ 5 5 √ x THINK AND DISCUSS 1 _ 10 1. ( √ 25 ) = 25 10 = 25 2 = √25 = 5 5 (xy ) 4b. _ xy 1. Explain how to find the value of ( √ 25 ) . 10 2. GET ORGANIZED Copy and complete the graphic organizer. In each cell, provide the definition and a numerical example of each type of exponent. 2. See p. A7. 5 Exponent Definition Numerical Example _1 bn m _ bn 490 Chapter 7 Exponents and Polynomials 3 Close A1NL11S_c07_0488_0493.indd 490 Summarize Essential Question Have students give an equivalent expression with fractional exponents for each expression below. Then have them simplify the expression. 4 1. √ 81 2. ( √ 25 ) 1 _ 81 4 ; 3 3. ( √ 256 ) 4 490 3 _ 3 25 2 ; 125 3 3 _ 256 4 ; 64 Chapter 7 Be sure students can answer the lesson’s essential question: How do you simplify an expression with a rational exponent? If the exponent has the 1 form __ n , find the nth root of the base. m If the exponent has the form __ n , find the nth root of the base raised to the mth power. and INTERVENTION Diagnose Before the Lesson 7-5 Warm Up, TE p. 488 Monitor During the Lesson Check It Out! Exercises, SE pp. 488–490 Questioning Strategies, TE pp. 489–490 Assess After the Lesson 7-5 Lesson Quiz, TE p. 493 Alternative Assessment, TE p. 493 6/25/09 3:18:00 PM 7-5 Exercises 7-5 Exercises KEYWORD: MA11 7-5 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide 5 1. Vocabulary In the expression √ 3x, what is the index? 5 Assign Guided Practice exercises as necessary. Simplify each expression. 1 _ SEE EXAMPLE 1 3. 16 2 4 6. 81 2 9 1 _ 1 _ 1 _ 2 1 _ 1 _ 1 _ 12. 81 4 + 8 3 5 3 _ 17. 25 2 125 2 _ 3 _ 19. 64 3 256 1 _ 13. 25 2 - 1 4 4 16. 125 3 25 4 _ 18. 36 2 216 9. 625 4 5 2 _ 15. 8 3 32 3 _ p. 489 1 _ 5 _ 14. 81 4 27 1 _ 8. 1 9 1 11. 8 3 + 64 2 10 3 _ SEE EXAMPLE 5. 27 3 3 1 _ 7. 216 3 6 10. 36 2 + 1 3 7 1 _ 4. 0 6 0 1 _ 1 _ p. 488 1 _ 1 _ 2. 8 3 2 20. 1 4 1 21. 0 3 0 1 _ SEE EXAMPLE 3 p. 489 SEE EXAMPLE 4 p. 490 22. Geometry Given a square with area a, you can use the formula P = 4a 2 to find the perimeter P of the square. Find the perimeter of a square that has an area of 64 m 2. 32 m Simplify. All variables represent nonnegative numbers. 23. 27. x 4y 2 x 2 y √ (a ) √ a a 1 2 _ 2 2 4 24. √z4 z 2 28. 25. (x ) √y x y 1 6 _ 4 3 2 4 x 6y 6 x 3y 3 √ 3 12 6 26. √a b a 4b 2 ( ) 1 3 _ z3 x 6y 9 3 √ 30. _ y x2 3 29. _ 1 √ z2 Quickly check key concepts. Exercises: 34, 40, 44, 51, 52, 56 31–42 43–50 51 52–59 1 2 3 4 Simplify each expression. 1 _ 32. 1 5 1 1 _ 1 _ 3 _ Extra Practice 1 _ 40. 25 2 - 81 4 2 44. 27 3 9 38. 400 2 20 1 _ 1 _ 41. 121 2 - 243 5 8 3 _ 45. 256 4 64 5 _ 3 _ 47. 100 2 1000 1 _ 37. 256 8 2 2 _ 43. 4 2 8 Skills Practice p. S17 1 _ 34. 729 2 27 1 _ 36. 196 2 14 39. 125 3 + 81 2 14 1 _ 33. 512 3 8 1 _ 35. 32 5 2 1 _ 1 _ 1 _ 31. 100 2 10 5 _ 48. 1 3 1 If you finished Examples 1–4 Basic 32–58 even, 60–80, 85, 87–90, 98–107 Average 32–58 even, 60–90, 92–96 even, 98–107 Advanced 32–80 even, 81–107 Homework Quick Check PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example If you finished Examples 1–2 Basic 31–50, 87, 88 Average 32–50 even, 60–78 even, 98–107 Advanced 32–50 even, 60–78 even, 86–88, 91–93, 98–107 1 _ 1 _ 42. 256 4 + 0 3 4 5 _ 46. 64 6 32 2 _ 49. 9 2 243 50. 243 5 9 Application Practice p. S34 51.. Biology Biologists use a formula to estimate the mass of a mammal’s brain. For a mammal with a mass of m grams, the approximate mass B of the brain, also in 2 _ grams, is given by B = __18 m 3 . Find the approximate mass of the brain of a mouse that has a mass of 64 grams. 2g Simplify. All variables represent nonnegative numbers. 3 6 9 52. √a c a 2c 3 56. 3 3 2m 53. √8m ( ) √x x y 2 1 _ x 2 y3 2 2 6 57. 1 3 _ a 2b 4 2 ( ) √ b 6 ab 4 54. 58. 4 x 16y 4 x 4y √ 3 x 6y 6 √ _ yx 2 3 6 3x 2 55. √27x (a b ) _ 2 y 59. 1 4 _ 2 √ b2 a 8b Fill in the boxes to make each statement true. _ 60. 256 4 = 4 1 64. _ 64 3 = 16 2 61. 1 _ 5 65. 3 _ 4 =11 = 125 625 1 _ 62. 225 = 15 2 4 _ 66. 27 = 81 3 63. 1 _ 6 67. _ 36 2 =0 0 = 216 3 7- 5 Rational Exponents A1NL11S_c07_0488_0493.indd 491 491 7/18/09 4:53:46 PM KEYWORD: MA7 Resources Lesson 7-5 491 Exercise 85 involves using rational exponents to model a real-world situation related to light. This exercise prepares students for the Multi-Step Test Prep on page 494. Simplify each expression. 81 _12 9 68. _ 169 13 ( ) _ 1 _1 71. (_ 16 ) 4 8 16 _ 74. (_ 81 ) 27 1 1 _ 77. (_ 81 ) 27 1 _ 2 3 _ 4 3 _ 1 3·_ 2 3 _ 42 1 _ ·3 42 1 _ 1 _ = (4 3) 2 = 64 2 = 8; = (4 ) 1 _ 2 1 _ 3 _2 256 _4 (_ 81 ) 3 8 _4 73. (_ 27 ) 9 8 _ 4 76. (_ 25 ) 125 16 8 _ 79. (_ 125 ) 625 3 27 _ 3 _ 2 8 _ 2 _ 3 64 3 _ 2 343 9 _ 4 _ 3 16 80. Multi-Step Scientists have found that the life span of a mammal living in captivity is related to the mammal’s mass. The life span in years L can be approximated by the formula 3 = = 2 = 8. It is often easier to take the square root first so that the remaining numbers in the calculation are smaller. 3 Typical Mass of Mammals Mammal L= where m is the mammal’s mass in kilograms. How much longer is the life span of a lion compared with that of a wolf? 12 years Mass (kg) Koala 8 Wolf 32 1 _ 12m 5 , 101. n < 3 1 _ 4 70. 3 _ 2 2 _ 3 3 _ 4 Answers 86. 4 2 = 4 8 (_ 27 ) 9 72. (_ 16 ) 4 75. (_ 49 ) 27 78. (_ 64 ) 69. Lion 243 Giraffe 1024 1 _ 81. Geometry Given a sphere with volume V, the formula r = 0.62V 3 may be used to approximate the sphere’s radius r. Find the approximate radius of a sphere that has a volume of 27 in 3. 1.86 in. 102. x ≥ 2 103. y ≤ - 2 104. D: {2}; R: {3, 4, 5, 6}; no; the domain value 2 is paired with several different range values. ( _) 1 3 3 82. Critical Thinking Show that a number raised to the power _13_ is the same as the cube root of that number. (Hint: Use properties of exponents to find the cube _1 · 3 1 _ 82. b = b3 = b 1 = b. Also, by 3 3 definition ( √ b) = 3 . Use the fact that if two numbers have of b 3 . Then compare this with the cube of √b the same cube, then they are equal.) 2 _ 3 _ Thinking Compare n 3 and n 2 for values of n greater than 1. When _1 383. Critical simplifying each of these expressions, will the result be greater than n or less than b. b. Therefore, b 3 = √ n? Explain. _2 3 _ 83. n 3 will be less 84. /////ERROR ANALYSIS///// Two students simplified 64 2 . Which solution is incorrect? than n because Explain the error. __2 < 1. n _2 will be greater 3 than n because __3 > 1. 3 105. D: {-2, -1, -0, 1}; R: {0, 1, 2, 3}; yes; each domain value is paired with exactly one range value. ! ?, 2 106. D: {5, 7, 9, 11}; R: {2}; yes; each domain value is paired with exactly one range value. 84. Solution A is incorrect. The first line should be _3 ! , " /- /-+ Ȗе !-" + */ " + ?, " ! /-+ Ȗе /- !1", .*+ , 64 2 = ( √ 64 ) . 3 107. D: 1 ≤ x ≤ 4; R: 2 ≤ y ≤ 4; yes; each domain value is paired with exactly one range value. 85. This problem will prepare you for the Multi-Step Test Prep on page 494. You can estimate an object’s distance in inches from a light source by using the ( ) _1_ L 2 , where L is the light’s luminosity in lumens and B is the light’s formula d = 0.8__ B brightness in lumens per square inch. a. Find an object’s distance to a light source with a luminosity of 4000 lumens and a brightness of 32 lumens per square inch. 10 in. b. Suppose the brightness of this light source decreases to 8 lumens per square inch. How does the object’s distance from the source change? 7-5 PRACTICE A The distance doubles (20 in.) 7-5 PRACTICE C a e ________________________________________ Practice B LESSON 7-5 a e __________________ C ass__________________ 7-5 PRACTICE B 492 Rational Exponents Chapter 7 Exponents and Polynomials Simplify each expression. All variables represent nonnegative numbers. ________________________________________ 1 1 1 1. 27 3 2. 1212 3. 0 3 3 4. 11 1 64 2 + 1 27 3 5. 11 1 16 4 1 + 83 6. 4 1 8. 25 8 8 5 3 11. 16 12. 1212 5 1 y5 14. 1 A number raised to the power of 1 is equal to the n 92 = 9 = 3 In general, if b > 1 and n is an integer where n ≥ 2, 1 32 5 then b n = n b . 1 nth root of that number. = 32 = 2 15. 3 Definition m of b n 1 1 17. ( x 3 y )3 x 2 y 2 x5 x 2y 4 18. x3 A number raised to 3 3 m 16 4 = 4 16 = 23 = 8 is the power of n 3 3 equal to the nth 100 2 = 100 root of the number raised to the mth = 103 = 1000 power. ( ) In general, if b > 1 and m and n are integers where m ≥ 1 and n ≥ 2, then m ) bn = (n b ) m n = bm . 1. The expression 1 27 3 is equal to the 3rd or cube root of 27. 4. 7. 4 32 5 is equal to the 5th root of 32 raised to the 4th power. 5. d h t th i i l t t th ibilit f th i t Chapter 7 1 812 1 + 32 5 1 = 81 + 5 32 =9+2 = 11 1 1 1 1. 64 2 2. 1000 3 3. 15 8 10 6. 9 2 1 400 2 2 32 5 3 814 8. 9. 4 1 1 1 5. 32 5 6. 49 2 4 2 7 1 27 5 3 11. 16 2 12. 256 4 l 1024 t tC i ht © b H lt M D l Additi d h 64 t th i i l t t th ibilit f th i t t 1 1 1 8. 1212 + 27 3 1 3 1 1 11. 144 2 − 125 3 7 1 9. 32 5 + 12 14 1 1 3 Oi i 1 10. 814 − 16 4 10. 4 2 t 1 4. 256 4 6 1 16 4 2 8 492 1 1 812 7. 8 3 + 16 2 1 83 20 l Additi 216 = 6. Simplify 812 + 32 5 . 5 2. The expression 9 2 is equal to the square root of 9 raised to the 5th power. i ht © b H lt M D 7/20/09 4:52:27 PM 3 Simplify each expression. 3. The expression tC 1 63 = 6 × 6 × 6 = 216, so Simplify each expression. t 1 , write the nth root of the number. n 1 216 3 . Think: What number, when taken as a factor 3 times, is equal to 216? Complete each of the following. 20 m l Simplify 1 x 1 19. Given a cube with volume V, you can use the formula P = 4V 3 to find the perimeter of one of the cube’s square faces. Find the perimeter of a face of a cube that has volume 125 m3. Oi i Rational Exponents 125 3 = 3 125 = 5 ( (x 4 )8 3 7-5 RETEACH 7-5 When an expression contains two or more expressions with fractional exponents, evaluate the expressions with the exponents first, then add or subtract. a 6 b3 a 2b 16. ( x 2 )4 x 6 Review for Mastery LESSON 216 3 = 3216 = 6 1 5 Name ________________________________________ Date __________________ Class__________________ To simplify a number raised to the power of 1331 x 4 y 12 x 2y 6 y 1 bn A1NL11S_c07_0488_0493.indd 492 3 8 The table below summarizes the definitions you need when you work with fractional exponents. of 9. 32 5 125 Use a Table Definition 3 3 2 10. 16 4 13. − 1 64 6 8 1 7. 15 + 49 2 1 100 2 __________________ 7-5 READING STRATEGIES 7-5 0 __________________ Reading Strategies LESSON 1 1 12. 625 4 − 0 2 5 3 _ 1 3·_ 1 ·3 _ 86. Write About It You can write 4 2 as 4 2 or as 4 2 . Use the Power of a Power Property to show that both expressions are equal. Is one method easier than the other? Explain. "" In Exercise 80, students might multiply by 12 before evaluating the fifth root of m. Remind students of the order of operations: grouping symbols, exponents, multiply/divide, add/subtract. Remind them that radical signs are grouping symbols. 1 _ 1 _ 87. What is 9 2 + 8 3 ? 4 5 6 10 88. Which expression is equal to 8? 3 _ 1 _ 4 _ 3 _ 42 16 2 32 5 64 2 a 3b a 3b 3 3 9 3 89. Which expression is equivalent to √a b ? a 2b a3 If students answer B for Exercise 89, they may think that 3 _ 1 _ 90. Which of the following is NOT equal to 16 2 ? ) ( √16 3 ) ( √16 2 3 43 3 √ b 3 = 1 or (b 3) 3 = 1. Remind 3 √16 them that cubing a number and taking the cube root are inverse operations that “undo” each other, 1 _ 3 b 3 = (b 3) 3 = b. so √ CHALLENGE AND EXTEND Use properties of exponents to simplify each expression. (a )(a )(a ) a 1 _ 3 91. 1 _ 3 1 _ 3 92. (x ) (x ) x 1 5 _ 2 3 _ 2 Ê,,", ,/ 4 93. (x ) (x ) 1 _ 5 3 1 4 _ 3 x3 You can use properties of exponents to help you solve equations. For example, to _1 _1 1 power to get (x 3) 3 = 64 3 . Simplifying both solve x 3 = 64, raise both sides to the __ 3 sides gives x = 4. Use this method to solve each equation. Check your answer. 1 x3 2 94. y 5 = 32 2 95. 27x 3 = 729 3 96. 1 = _ 8 97. Geometry The formula for the surface area of a sphere S in terms of its volume V is 1 _ 3 2 _ 3 . What S = (4π) (3V ) is the surface area of a sphere that has a volume of 36π cm ? Leave the symbol π in your answer. What do you notice? 36π cm 2; both volume and 3 surface area are described by 36π (although the units are different). Journal Have students write the steps they 5 _ would use to simplify 729 6 . SPIRAL REVIEW Solve each equation. (Lesson 2-6) 98. ⎪x + 6⎥ = 2 -8, -4 100. ⎪2x - 1⎥ = 3 -1, 2 99. ⎪5x + 5⎥ = 0 -1 Have students write a quiz on rational exponents. The quiz should include five problems about simplifying expressions of various types. The quiz should also include answers. Solve each inequality and graph the solutions. (Lesson 3-4) 1x + 3 101. 3n + 5 < 14 102. 4 ≤ _ 103. 7 ≥ 2y + 11 2 Give the domain and range of each relation. Tell whether the relation is a function. Explain. (Lesson 4-2) 104. {(2, 3), (2, 4), (2, 5), (2, 6)} 105. {(-2, 0), (-1, 1), (0, 2), (1, 3)} 106. 107. x y 5 2 7 2 9 Ó Simplify each expression. ä 1. 16 4 1 _ Ó 2 11 7-5 { Ó { Ó 2 7- 5 Rational Exponents _______________________________________ __________________ __________________ Problem Solving 7-5 PROBLEM SOLVING LESSON 7-5 Rational Exponents Write the correct answer. 1. For a pendulum with a length of L meters, the time in seconds that it takes the pendulum to swing back and forth is 2. The Beaufort Scale is used to measure the intensity of tornados. For a tornado with Beaufort number B, the formula 1 Challenge LESSON 7-5 7-5 CHALLENGE 6s 4. 64 START 7/18/09 4:53:59 PM 1 16 4 51.3 mi/h 1 42 − 1 27 3 1 2 05 1 2 13 − 8 3 216 3 1 3 83 + 92 4. At a factory that makes cylindrical cans, 1 V 2 the formula r = is used to find the 12 3 A2 . by the formula V = Find the volume of a cube whose faces each have an area of 64 in2. 1 1 36 2 − 216 3 radius of a can with volume V. What is the radius of a can whose volume is 192 cm3? 1 4 cm 512 in3 1 _ - 27 3 9 729 7 _ 6 128 5. In an experiment, the approximate population P of a bacteria colony is given by Keep Growing! Find a path from start to finish in the maze below. Each box that you pass through must have a value that is greater than or equal to the value in the previous box. You may only move horizontally or vertically to go from one box to the next. 3 3. Given a cube whose faces each have area A, the volume of the cube is given 3. 3 _ 81 2 2 Name ________________________________________ Date __________________ Class__________________ v = 1.9B 2 may be used to estimate the tornado’s wind speed in miles per hour. Estimate the wind speed of a tornado with Beaufort number 9. approximately 2L2 . About how long A1NL11S_c07_0488_0493.indd does 493it take a pendulum that is 9 meters long to swing back and forth? 493 2. 1 _ 144 2 1 2 1 25 2 − 32 5 1 3 1 92 − 83 15 64 6 3 2 3 2 12 − 9 2 125 3 814 1000 3 1 1 2 100 2 + 27 3 1 1 16 2 − 16 4 3 32 5 + 0 4 5 _ P = 15t 3 , where t is the number of days since the start of the experiment. Find the population of the colony on the 8th day. 480 3 Given an animal’s body mass m, in grams, the formula B = 1.8m 4 may be used to estimate the mass B, in grams, of the animal’s brain. The table shows the body mass of several birds. Use the table for questions 5–7. Select the best answer. 5. Which is the best estimate for the brain mass of a macaw? A 9g C 125 g B 45 g D 6. How much larger is the brain mass of a barn owl compared to the brain mass of a cockatiel? F 189 g 225 g G 7. An animal has a body mass given by the expression x 4 . Which expression can be used to estimate the animal’s brain mass? A B = 1.8x3 340.2 g 2 1 1 32 5 27 3 1210 + 12 1 3 16 4 3 1 16 4 − 12 1 102410 1 1 Simplify. All variables represent nonnegative numbers. 9 2 − 30 1 4 814 + 49 2 243 5 1 1 1 2 16 4 + 32 5 3 100 2 625 4 128 7 1 1 144 2 − 812 1 64 3 5 6. √ x 10z 5 Body Mass (g) Cockatiel 81 Guam Rail 256 D B = 1.8x 1 49 2 + 0 2 J 1215 g Bird C B = 1.8x12 1 4 92 + 34 H 388.8 g Typical Body Masses of Birds 3 B B = 1.8x 4 2 512 9 Macaw 625 Barn Owl 1296 1 3 125 3 − 20 1 625 4 4 1 3 2 814 − 32 5 256 2 16 4 + 4 2 64 3 32 5 + 100 2 2 243 5 2 125 3 3 128 7 1 243 5 5 64 6 2 − 125 3 3 1 1 7. ( ) 1 43 _ a 4b 4 x 2z √ b3 a 16b 2 Also available on transparency Sources: http://www.beyondveg.com/billings-t/companat/comp-anat-appx2.shtml http://www.sandiegozoo.org/animalbytes/index.html Oi i l t tC i ht © b H lt M D l Additi d h t th i i l t t th ibilit f th i t t FINISH Oi i l t tC i ht © b H lt M D l Additi d h t th i i l t t th ibilit f th i t t Lesson 7-5 493 SECTION 7A SECTION 7A Exponents I See the Light! The speed of light is the product of its frequency f and its wavelength w. In air, the speed of light is 3 × 10 8 3 × 10 8 m/s. 8 ) -1 ( Organizer then solve this equation for frequency. Write this equation as an equation with w raised to a negative exponent. Objective: Assess students’ GI ability to apply concepts and skills in Lessons 7-1 through 7-5 in a real-world format. <D @<I _; f = 3 × 10 w w 1. Write an equation for the relationship described above, and 1. f = 2. Wavelengths of visible light range from 400 to 700 nanometers (10 -9 meters). Use a graphing calculator and the relationship you found in Problem 1 to graph frequency as a function of wavelength. Sketch the graph with the axes clearly labeled. Describe your graph. Online Edition 3. The speed of light in water is __34 of its speed in air. Find the speed of light in water. 2.25 × 10 8 m/s 4. When light enters water, some colors bend more than others. How much the light bends depends on its wavelength. This is what creates a rainbow. The frequency of green light is about 5.9 × 10 14 cycles per second. Find the wavelength of green light in water. about 3.81 × 10 -7 m Resources Algebra I Assessments 5. When light enters water, www.mathtekstoolkit.org Problem Text Reference 1 Lesson 7-1 2 Lesson 7-2 3 Lesson 7-3 4 Lesson 7-3 5 Lesson 7-4 colors with shorter wavelengths bend more than colors with longer wavelengths. Violet light has a frequency of 7.5 × 10 14 cycles per second, and red light has a frequency of 4.6 × 10 14 cycles per second. Which of these colors of light will bend more when it enters water? Justify your answer. Answers 2. Violet light will bend more. Students can justify by using the inverse relationship, by using the graph, or by finding the wavelength for each color and comparing them. As the wavelength increases, the frequency decreases. 494 Chapter 7 Exponents and Polynomials 5. How does wavelength change as frequency changes? As one increases, the other decreases. INTERVENTION Scaffolding Questions A1NL11S_c07_0494.indd 494 1. What does it mean to “solve for frequency”? Isolate the variable that represents frequency. 2. What are a reasonable domain and range? Possible answer: D: 380 to 760; R: 3 × 1014 to 9 × 1014 3. How do you enter numbers in scientific notation into a calculator? Possible answer: 3 × 1014 is entered as 3 KEYWORD: MA7 Resources 494 Chapter 7 14 or 3 14. 4. What information do you need for this problem? speed of light in water Where can you find this information? answer to Problem 3 6/25/09 9:15:23 AM Extension When you choose an FM radio station, you are choosing the frequency in MHz, or millions of waves per second (90 MHz = 90,000,000 waves per second). Find the wavelength for your favorite FM radio station. Possible answer: Using w = (3 × 108) f -1, FM 93.7 has an approximate wavelength of 3.2 m. SECTION 7A SECTION Quiz for Lessons 7-1 Through 7-5 7A 7-1 Integer Exponents Evaluate each expression for the given value(s) of the variable(s). _ _ 2. n -3 for n = -5 - 1 1. t -6 for t = 2 1 Simplify. 5 _ 4. 5k -3 k 3. r 0 s -2 for r = 8 and s = 10 125 64 x4 5. _ y -6 3 6. 8f -4 g 0 x 4y 6 8. Measurement Metric units can be written in terms of a base unit. The table shows some of these equivalencies. Simplify each expression. _8 f a -3 7. _ b -2 4 1 _ 100 b _ Objective: Assess students’ a3 mastery of concepts and skills in Lessons 7-1 through 7-5. 2 Selected Metric Prefixes Milli- Centi- Deci- 10 -3 10 -2 10 -1 Deka- Hecto10 1 Organizer Kilo- 10 2 10 3 Resources 7-2 Powers of 10 and Scientific Notation Assessment Resources 10. Write 0.0000001 as a power of 10. 10 -7 12. Find the value of 82.1 × 10 4. 821,000 9. Find the value of 10 4. 10,000 11. Write 100,000,000,000 as a power of 10. 10 11 Section 7A Quiz 13. Measurement The lead in a mechanical pencil has a diameter of 0.5 mm. Write this number in scientific notation. 5 × 10 -1 Test & Practice Generator 7-3 Multiplication Properties of Exponents Simplify. 15. 3 5 · 3 -3 3 2 , or 9 14. 2 2 · 2 5 2 7 INTERVENTION 1 17. a 3 · a -6 · a -2 _ 16. p 4 · p 5 p 9 a5 18. Biology A swarm of locusts was estimated to contain 2.8 × 10 10 individual insects. If each locust weighs about 2.5 grams, how much did this entire swarm weigh? Write your answer in scientific notation. 7 × 10 10 g Resources Ready to Go On? Intervention and Enrichment Worksheets Simplify. 19. (3x 4 )3 27x 12 20. (m 3 n 2 )5 m 15 n 10 21. (-4d 7 )2 16d 14 22. (cd 6 )3 · (c 5 d 2 )2 Ready to Go On? CD-ROM c 13 d 22 7-4 Division Properties of Exponents Simplify. 69 23. _ 36 67 12a 5 4a 3 24. _ 3a 2 25. 27 () _ 125 3 _ 5 3 26. ( ) 4p 3 _ 2pq 4 2 4p _ 4 q8 Answers Simplify each quotient and write the answer in scientific notation. 27. (8 × 10 9 ) ÷ (2 × 10 6 ) 4 × 10 (3.5 × 10 5 ) ÷ (7 × 10 8 ) 28. 5 × 10 3 -4 29. (1 × 10 4 ) ÷ (4 × 10 4 ) 2.5 × 10 -1 7-5 Rational Exponents Simplify each expression. All variables represent nonnegative numbers. 1 _ 1 _ 30. 81 2 9 8 y4 34. √x x 4y 2 3 _ 31. 125 3 5 32. 4 2 8 3 35. √r 9 6 36. √z12 ___ ___ r3 2 _ 33. 0 9 0 z2 ______ 37. √p 3 q12 pq 4 3 Ready to Go On? NO A1NL11S_c07_0495.indd 495 READY Ready to Go On? Intervention TO Worksheets 103 = 1000 495 YES Diagnose and Prescribe INTERVENE 1 8. 10-3 = _, or 0.001; 1000 1 10-2 = _, or 0.01; 100 1 10-1 = _, or 0.1; 10 101 = 10; 102 = 100; ENRICH 7/20/09 5:03:23 PM GO ON? Intervention, Section 7A CD-ROM Lesson 7-1 7-1 Intervention Activity 7-1 Lesson 7-2 7-2 Intervention Activity 7-2 Lesson 7-3 7-3 Intervention Activity 7-3 Lesson 7-4 7-4 Intervention Activity 7-4 Lesson 7-5 7-5 Intervention Activity 7-5 Online Diagnose and Prescribe Onlinew READY TO GO ON? Enrichment, Section 7A Worksheets CD-ROM Online Ready to Go On? 495 SECTION 7B Polynomials One-Minute Section Planner Lesson Lab Resources Lesson 7-6 Polynomials • • □ Optional Classify polynomials and write polynomials in standard form. Evaluate polynomial expressions. SAT-10 ✔ NAEP ✔ ACT ✔ SAT ✔ SAT Subject Tests □ □ □ Use algebra tiles to model polynomial addition and subtraction. SAT-10 ✔ NAEP ACT SAT SAT Subject Tests □ □ □ □ Lesson 7-7 Adding and Subtracting Polynomials • □ Add and subtract polynomials. ✔ SAT-10 ✔ NAEP ✔ ACT □ □ □ index cards, scissors, tape, ruler (MK), 8.5-by-11 inch paper □ 7-7 Algebra Lab Model Polynomial Addition and Subtraction • □ Materials SAT □ Algebra Lab Activities 7-7 Lab Recording Sheet Required Technology Lab Activities 7-7 Technology Lab Optional Algebra Lab Activities 7-8 Lab Recording Sheet Required algebra tiles (MK) books, pencils, slips of paper SAT Subject Tests 7-8 Algebra Lab Model Polynomial Multiplication • Use algebra tiles to model polynomial multiplication. □ SAT-10 □ NAEP □ ACT □ SAT □ SAT Subject Tests algebra tiles (MK) Lesson 7-8 Multiplying Polynomials • Multiply polynomials. ✔ NAEP □ ✔ ACT □ SAT-10 □ □ SAT □ SAT Subject Tests Lesson 7-9 Special Products of Binomials • Find special products of binomials ✔ NAEP □ ✔ ACT □ SAT-10 □ □ SAT □ SAT Subject Tests Note: If NAEP is checked, the content is tested on either the Grade 8 or Grade 12 NAEP assessment. 496A Chapter 7 MK = Manipulatives Kit Math Background TERMINOLOGY Lesson 7-6 POLYNOMIAL OPERATIONS Lessons 7-7 to 7-9 In order to discuss polynomials, we must agree on terminology. The basic unit is the monomial. A monomial is a product of a real number and one or more variables with whole-number exponents. (The real number is usually rational, particularly within the scope of Algebra 1, but this is not a requirement.) A polynomial is a sum of monomials. For example, the polynomial 8x 4 - 3x - 1 may be written as 8x 4 + (-3x) + (-1), which is the sum of the monomials 8x 4, -3x, and -1. Adding and subtracting polynomials is fairly straightforward because the process is nothing more than combining like terms. POLYNOMIALS Lesson 7-6 Polynomials are in many ways analogous to counting numbers. Because our number system is base 10, all counting numbers can be written in expanded form in terms of powers of 10. For example, consider the expanded form of 653. 653 = 6 · 100 + 5 · 10 + 3 · 1 = 6 · 10 2 + 5 · 10 1 + 3 · 10 0 You can create a polynomial by replacing each of the 10s by a variable, such as x. 6 · 10 2 + 5 · 10 1 + 3 · 10 0 6 · x2 + 5 · x1 + 3 · x0 This polynomial is usually written in the more familiar form 6x 2 + 5x + 3. For counting numbers, the only permissible multipliers of the powers of 10 are the digits 0 through 9, inclusive. For polynomials, any real number can be a multiplier of the variable terms. The goal of this analogy is not to suggest that there is a correspondence between counting numbers and polynomials but to demonstrate that diverse mathematical concepts sometimes share underlying structures. As such, it makes sense to pose some of the same questions about polynomials that one might pose about counting numbers. For example, can we add, subtract, multiply, and divide polynomials? How? (3x 2 + 7x + 5) + (2x + 6) = 3x 2 + 9x + 11 Polynomial multiplication can present greater difficulty for students, so it is essential to build gradually. Multiplication of two monomials is a natural starting point. You can use the Commutative and Associative Properties to show that (4x)(2x) = 8x 2. Use the Distributive Property when multiplying a monomial and a binomial: 5x(2x + 3) = (5x)(2x) + (5x)(3) = 10x 2 + 15x The Distributive Property is used repeatedly when multiplying a binomial by a binomial: (3x + 2)(7x + 4) = (3x)(7x + 4) + (2)(7x + 4) = 21x 2 + 12x + 14x + 8 = 21x 2 + 26x + 8 In fact, the Distributive Property can be used to multiply any two polynomials, regardless of the number of terms. The product will have one term for each product of a term from the first polynomial and a term from the second polynomial. So, the product of a binomial (2 terms) and a trinomial (3 terms) will have 2 · 3 = 6 terms before simplifying: (x + 2)(x 2 + 6x + 8) = (x + 2)(x 2) + (x + 2)(6x) + (x + 2)(8) = (x)(x 2) + 2(x 2) + x(6x) + 2(6x) + x(8) + 2(8) It is important to remember that this rule is true before the product is simplified. Clearly, some of the terms above are like terms and will be combined; the final answer will have fewer than 6 terms. In general, the product of a polynomial with m terms and a polynomial with n terms has mn terms before simplifying. 496B 7-6 Organizer 7-6 Pacing: Traditional 1 day Block Polynomials __1 day 2 Objectives: Classify polynomials and write polynomials in standard form. GI Evaluate polynomial expressions. @<I <D Evaluate polynomial expressions. Online Edition Tutorial Videos Countdown Week 16 Warm Up Evaluate each expression for the given value of x. 1. 2x + 3; x = 2 Who uses this? Pyrotechnicians can use polynomials to plan complex fireworks displays. (See Example 5.) Objectives Classify polynomials and write polynomials in standard form. A monomial is a number, a variable, or a product of numbers and variables with wholenumber exponents. Vocabulary monomial degree of a monomial polynomial degree of a polynomial standard form of a polynomial leading coefficient quadratic cubic binomial trinomial Monomials EXAMPLE 1 Find the degree of each monomial. A -2a 2b 4 7. 2n7 4 B 4 69 2 6. y3 8. Add the exponents of the variables: 2 + 4 = 6 The degree is 6. 2 4x 0 The degree is 0. Identify the coefficient in each term. 5. 4x3 2 -0.3x -2 4x - y _ x3 Finding the Degree of a Monomial 13 3. -4x - 2; x = -1 4. 7x2 + 2x; x = 3 0.5x The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0. 7 2. x2 + 4; x = -3 -7xy 5 x Not Monomials 4 C 8y 1 1 -s4 There is no variable, but you can write 4 as 4x 0. 8y The degree is 1. -1 Also available on transparency Find the degree of each monomial. 1a. 1.5k 2m 3 1b. 4x 1 The terms of an expression are the parts being added or subtracted. See Lesson 1-7. Q: What happened to the quadratic polynomial when he fell asleep on the beach? A variable written without an exponent has exponent 1. 1c. 2c 3 3 A polynomial is a monomial or a sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree. EXAMPLE 2 Finding the Degree of a Polynomial Find the degree of each polynomial. A 4x - 18x 5 A: He got second-degree burns. 4x: degree 1 -18x 5: degree 5 Find the degree of each term. The degree of the polynomial is the greatest degree, 5. 496 Chapter 7 Exponents and Polynomials 1 Introduce A1NL11S_c07_0496-0501.indd e x p l o r496 at i o n 7-6 Motivate Polynomials 1. The table shows examples of expressions that are and are not polynomials. What are some characteristics that the polynomials have in common? Describe some ways in which the polynomials are different from the expressions that are not polynomials. Polynomials Not Polynomials 4x 2 ⫺ 5x ⫹ 1 3x 0.6x ⫺1 ⫹ 7 8 3y ⫺ 0.5y 4 __ ⫺9x 2 1 ⫺2z 3 ⫹ 3z 2 ⫹ __ z 1.7xy 3xy 4 x2 3 ⫺5 ⫹ 3x y 2 ⫺8 1 ⫹2 ___ 2 4x 2y ⫺ x 3y 2 xy 2. What do you notice about the exponents in the polynomials? 3. Do you think 16x ⫹ 2xy ⫹ 8y 2 KEYWORD: MA7 Resources ⫺2 Chapter 7 n: 2 p: 1 Ask students to identify the greatest exponent. 3 is a polynomial? Why or why not? THINK AND DISCUSS 4. Show your own examples of expressions that are and are not polynomials. 5. Describe how you can tell whether an expression is a polynomial. 496 Then have students identify the exponent of each variable. m: 3 3z ⫹ __ 1 1 z 5 ⫺ 2z 4 ⫹ 8z 2 ⫹ __ __ 2 Write m3 + n2 + p + 5 on the board. Ask students how many terms are in the expression. 4 Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 6/25/09 9:21:24 AM Find the degree of each polynomial. "" B 0.5x y + 0.25xy + 0.75 0.5x 2y : degree 3 0.25xy : degree 2 0.75: degree 0 The degree of the polynomial is the greatest degree, 3. Students often confuse the degree of a polynomial with the number of terms. Students may be less likely to confuse the two if they equate the word degree with an exponentrelated phrase such as “maximum power.” C 6x 4 + 9x 2 - x + 3 9x 2: degree 2 -x: degree 1 6x 4: degree 4 The degree of the polynomial is the greatest degree, 4. 3: degree 0 Find the degree of each polynomial. 2a. 5x - 6 1 2b. x 3y 2 + x 2y 3 - x 4 + 2 5 The terms of a polynomial may be written in any order. However, polynomials that contain only one variable are usually written in standard form. The standard form of a polynomial that contains one variable is written with the terms in order from greatest degree to least degree. When written in standard form, the coefficient of the first term is called the leading coefficient . EXAMPLE 3 Find the degree of each monomial. Write each polynomial in standard form. Then give the leading coefficient. B. 7ed C. 3 Find the degree of each term. Then arrange them in descending order. 1 3 0 - 4x 3 - x 2 + 20x + 2 2 3 2 1 0 The standard form is -4x 3 - x 2 + 20x + 2. The leading coefficient is -4. B y 3 + y 5 + 4y Degree: 3 5 1 Example 3 3 1 Write each polynomial in standard form. Then give the leading coefficient. 3a. 16 - 4x 2 + x 5 + 9x 3 3b. 18y 5 - 3y 8 + 14y x + 9x - 4x + 16; 1 5 3 2 -3y 8 + 18y + 14y ; -3 5 Some polynomials have special names based on their degree and the number of terms they have. Degree Name Terms Name 0 Constant 1 Monomial 1 Linear 2 Binomial 2 Quadratic 3 Trinomial 3 Cubic 4 or more Polynomial 4 Quartic 5 Quintic 6 or more 6th degree, 7th degree, and so on 497 4 Write each polynomial in standard form. Then give the leading coefficient. A. 6x - 7x5 + 4x2 + 9 -7x5 + 4x2 + 6x + 9; -7 B. y2 + y6 - 3y y6 + y2 - 3y; 1 INTERVENTION Questioning Strategies EX AM P LE 7- 6 Polynomials 1 • If a monomial has more than one variable with an exponent, how do you determine its degree? • Why is the degree of a constant always zero? 2 Teach EX AM P LE A1NL11S_c07_0496-0501.indd 497 2 6/25/09 9:21:35 AM Guided Instruction Discuss with students examples and nonexamples of monomials. Explain how to find the degree of a monomial and the degree of a polynomial. Be sure students get plenty of exposure to the new vocabulary in this lesson—it will be used often throughout the remainder of this chapter and Chapter 8. Find the degree of each polynomial. 5 The standard form is y 5 + y 3 + 4y. The leading coefficient is 1. 5 0 y 5 + y 3 + 4y { ⎧ ⎨ ⎩⎧ ⎨ ⎩ { ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ y 3 + y 5 + 4y 2 A. 11x7 + 3x3 7 1 1 B. _ w2z + _ z 4 - 5 3 2 Find the degree of each term. Then arrange them in descending order. A variable written without a coefficient has a coefficient of 1. 7 Example 2 ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ 20x - 4x 3 + 2 - x 2 y = 1y Example 1 A. 4p4q3 A 20x - 4x 3 + 2 - x 2 5 Additional Examples Writing Polynomials in Standard Form Degree: Ê,,", ,/ 2 Through Kinesthetic Experience Divide students into groups of five, and have each write a monomial on an index card in large print. They may write a constant or a monomial with a variable. Tell them the variable must be x, but it can be raised to any power between 1 and 4, and can have any coefficient. Then call out classifications such as quadratic trinomial and quartic binomial. Have students in each group stand and arrange themselves to form that polynomial, holding their monomial in front of them for the rest of the class to see. • Do coefficients affect the degree of the polynomial? Explain. EX AM P LE 3 • When is the coefficient of the first term also the leading coefficient of the polynomial? Lesson 7-6 497 EXAMPLE 4 Classifying Polynomials Classify each polynomial according to its degree and number of terms. Additional Examples A 5x - 6 Degree: 1 Example 4 A. + 4n Degree: 2 Terms: 3 y 2 + y + 4 is a quadratic trinomial. C 6x 7 + 9x 2 - x + 3 cubic binomial Degree: 7 B. 4y6 - 5y3 + 2y - 9 6th-degree polynomial C. -2x 5x - 6 is a linear binomial. B y +y+4 Classify each polynomial according to its degree and number of terms. 5n3 Terms: 2 2 Terms: 4 6x 7 + 9x 2 - x + 3 is a 7th-degree polynomial. Classify each polynomial according to its degree and number of terms. 4a. x 3 + x 2 - x + 2 4b. 6 4c. -3y 8 + 18y 5 + 14y linear monomial cubic polynomial constant monomial 8th-degree trinomial Example 5 EXAMPLE A tourist accidentally drops her lip balm off the Golden Gate Bridge. The bridge is 220 feet from the water of the bay. The height of the lip balm is given by the polynomial -16t 2 + 220, where t is time in seconds. How far above the water will the lip balm be after 3 seconds? 76 ft 5 Physics Application A firework is launched from a platform 6 feet above the ground at a speed of 200 feet per second. The firework has a 5-second fuse. The height of the firework in feet is given by the polynomial -16t 2 + 200t + 6, where t is the time in seconds. How high will the firework be when it explodes? ¶ Substitute the time for t to find the firework’s height. -16t 2 + 200t + 6 -16(5)2 + 200 (5) + 6 -16(25) + 200 (5) + 6 -400 + 1000 + 6 INTERVENTION ÈÊvÌ The time is 5 seconds. Evaluate the polynomial by using the order of operations. 606 Questioning Strategies When the firework explodes, it will be 606 feet above the ground. EX A M P L E 4 5. What if…? Another firework with a 5-second fuse is launched from the same platform at a speed of 400 feet per second. Its height is given by -16t 2 + 400t + 6. How high will this firework be when it explodes? 1606 ft • Why will a constant polynomial always be a monomial? • How does writing a polynomial in standard form help you classify the polynomial? EX A M P L E THINK AND DISCUSS 5 • What step in the order of operations will never be used when evaluating a polynomial in standard form? Why? 1. Explain why each expression is not a polynomial: 2x 2 + 3x -3; 1 - _ab_ . 2. GET ORGANIZED Copy and complete the graphic organizer. In each oval, write an example of the given type of polynomial. *Þ>Ã >Ã >Ã 498 Chapter 7 Exponents and Polynomials Answers to Think and Discuss 3 Close Summarize A1NL11S_c07_0496-0501.indd 498 Have students give examples of monomials, binomials, trinomials, and polynomials of fourth and fifth degree. Write their examples on the board as they say them. For each, have students state whether the polynomial is in standard form, and if not, rewrite it in standard form. and INTERVENTION Diagnose Before the Lesson 7-6 Warm Up, TE p. 496 Monitor During the Lesson Check It Out! Exercises, SE pp. 496–498 Questioning Strategies, TE pp. 497–498 Assess After the Lesson 7-6 Lesson Quiz, TE p. 501 Alternative Assessment, TE p. 501 498 Chapter 7 /À>Ã 1. Possible answer: 2x2 + 3x-3 contains an expression with a negative a exponent. 1 - __ contains a variable b within a denominator. 2. See p. A7. 6/25/09 9:21:38 AM 7-6 Exercises 7-6 Exercises KEYWORD: MA11 7-6 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide Vocabulary Match each polynomial on the left with its classification on the right. 1. 2x 3 + 6 d Assign Guided Practice exercises as necessary. a. quartic polynomial 2. 3x 3 + 4x 2 - 7 c b. quadratic polynomial 3. 5x - 2x + 3x - 6 a c. cubic trinomial 2 4 If you finished Examples 1–3 Basic 27–49 Average 27–49 Advanced 27–49 d. cubic binomial SEE EXAMPLE 1 SEE EXAMPLE Find the degree of each monomial. 5. -7xy 2 3 4. 10 6 0 p. 496 2 9. 0.75a 2b - 2a 3b 5 8 11. r + r - 5 3 3 SEE EXAMPLE 3 12. a + a - 2a 3 2 3 15. 9a - 8a 2 8 b 2 - 2b 2+ 5; 1 g 2 + 5g - 7; 1 5 p. 498 13. 3k 4 + k 3 - 2k 2 + k 4 16. 5s - 3s + 3 - s If you finished Examples 1–5 Basic 27–78, 81–89 Average 27–79, 81–89 Advanced 27–74, 76–89 Homework Quick Check 7 -s 7 + 5s 24- 3s 3+ 3; -1 2 19. 3c + 5c + 5c - 4 Quickly check key concepts. Exercises: 27, 32, 34, 42, 52, 58, 60 5c 4 + 5c 3 + 3c 2 - 4; 5 Classify each polynomial according to its degree and number of terms. 21. x- 7 quadratic trinomial 2 3 linear binomial 2 3 quartic polynomial cubic binomial 23. q + 6 - q + 3q 4 SEE EXAMPLE 10. 15y - 84y 3 + 100 - 3y 2 3 2 18. 5g - 7 + g 3x 2 + 2x - 1; 3 20. x 2 + 2x + 3 p. 498 9 -8a 9 + 9a 28; -8 17. 2x + 3x - 1 SEE EXAMPLE 4 2 Write each polynomial in standard form. Then give the leading coefficient. 14. -2b + 5 + b p. 497 7. 2 0 Find the degree of each polynomial. 8. x 2 - 2x + 1 2 p. 496 6. 0.4n 8 8 24. 5k + 7k 26. Geometry The surface area of a cone is approximated by the polynomial 3.14r 2 + 3.14r, where r is the radius and is the slant height. Find the approximate surface area of this cone. 301.44 cm2 22. 8 + k + 5k 4 quartic trinomial 3 2 25. 2a + 4a - a 4 quartic trinomial ->ÌÊ i} ÌÊÊ£äÊV ÈÊV PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 27–34 35–40 41–49 50–57 58 1 2 3 4 5 Find the degree of each monomial. 27. 3y 4 4 28. 6k 1 29. 2a 3b 2c 6 30. 325 0 31. 2y 4z 3 7 32. 9m 5 5 33. p 1 34. 5 0 Find the degree of each polynomial. 35. a 2 + a 4 - 6a 4 38. -5f 4 + 2f 6 + 10f 8 8 36. 3 2b - 5 1 37. 3.5y 2 - 4.1y - 6 2 39. 4n 3 - 2n 3 40. 4r 3 + 4r 6 6 Extra Practice Skills Practice p. S17 Application Practice p. S34 Write each polynomial in standard form. Then give the leading coefficient. 41. 2.5 + 4.9t 3 - 4t 2 + t 42. 8a - 10a 2 + 2 43. x 7 - x + x 3 - x 5 + x 10 44. -m + 7 - 3m 45. 3x + 5x -2 4 + 5x 46. 2-2n + 1 - n 2 47. 4d + 3d - d + 5 48. 3s 2 + 12s + 6 49. 4x - x - x + 1 4.9t 3 - 4t 2 + t + 2.5; 4.9 2 7; 3-3 -3m - m + 2 2 -10a22 + 8a + 2; -10 3 5x 3 -d 3 + 3d 2 + 4d + 5; -1 x 10 + x 7 - x 5 + x 3 - x ; 1 + 3x3 + 5x - 4; 5 -n - 22n +5 1; -1 3 12s 3 + 3s 2 + 6; 12 -x 5 - x 3 + 4x 2 + 1; -1 7- 6 Polynomials 499 Teacher to Teacher For many students it is easier to identify the degree of each term in a polynomial when each term is written with the variables and their exponents. For example: A1NL11S_c07_0496-0501.indd 499 5x2y + 25xy + 75 = 5x2y1 + 25x1y1 + 75x0y0 5x2y1 degree 2 + 1 = 3 ⎯⎯ 25x1y1 degree 1 + 1 = 2 ⎯⎯ 75x0y0 degree 0 + 0 = 0 ⎯⎯ The highest degree is 3, so the polynomial 5x2y + 25xy + 75 has degree 3. 6/25/09 9:21:44 AM Arlane Frederick Buffalo, NY KEYWORD: MA7 Resources Lesson 7-6 499 Exercise 74 involves expressing the area and perimeter of a rectangle as polynomials. This exercise prepares students for the MultiStep Test Prep on page 528. Classify each polynomial according to its degree and number of terms. Transportation Kinesthetic Have students make (or attempt to make) the boxes for parts a, b, and c of Exercise 63. Give students scissors and tape and make sure they have a ruler and 8.5-by-11-inch paper (standard size of notebook or copy paper). The ruler can be found in the Manipulatives Kit (MK). Answers 78. Time (s) 86 3 81 4 44 52. 3.5x 3 - 4.1x - 6 53. 4g + 2g 2 - 3 54. 2x - 6x 55. 6 - s - 3s 56. c 2 + 7 - 2c 3 linear3 monomial 4 quadratic binomial quartic trinomial cubic trinomial quadratic trinomial 57. -y 2 cubic trinomial quadratic monomial Tell whether each statement is sometimes, always, or never true. Hybrid III is the crash test dummy used by the Insurance Institute for Highway Safety. During a crash test, sensors in the dummy’s head, neck, chest, legs, and feet measure and record forces. Engineers study this data to help design safer cars. 59 2 51. 6k constant monomial 2 58. Transportation The polynomial 3.675v + 0.096v 2 is used by transportation officials to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat, dry pavement. What is the stopping distance for a car traveling at 30 miles per hour? 196.65 ft Height (ft) 1 50. 12 59. A monomial is a polynomial. always 60. A trinomial is a 3rd-degree polynomial. sometimes 61. A binomial is a trinomial. never 62. A polynomial has two or more terms. sometimes 8.5 in. 63. Geometry A piece of 8.5-by-11-inch cardboard has identical squares cut from its corners. It is then folded into a box with no lid. The volume of the box in cubic inches is 4c 3 - 39c 2 + 93.5c, where c is the side length of the missing squares in inches. a. What is the volume of the box if c = 1 in.? 58.5 in3 b. What is the volume of the box if c = 1.5 in.? 66 c. What is the volume of the box if c = 4.25 in.? 0 11 in. in3 c d. Critical Thinking Does your answer to part c make sense? Explain why or why not. Yes; the width of the cardboard is The rocket will be highest after 2 s. 73. Possible answer: 8.5 in., so 4.25 in. cuts will meet, leaving nothing to fold up. First identify the Copy and complete the table by evaluating each polynomial for the given values of x. degree of each term. From left to right, Polynomial x = -2 x=0 x=5 the degrees are ) ( ) 5x - 6 5 -2 - 6 = -16 5(0 - 6 = -6 64. 3, 0, 2, 4, 19 5 3 and 1. x + x + 4x 65. -48 0 3270 Arrange the -10x 2 66. -40 -250 0 terms in order of decreasing degree, and move Give one example of each type of polynomial. Possible answers given. the plus or minus 67. quadratic trinomial 68. linear binomial 5x - 2 69. constant monomial 5 x 2 + 3x - 6 sign in front of each 3 70. cubic monomial 6x 71. quintic binomial x 5 - 3 72. 12th-degree trinomial term with it: 2x 12 - x + 15 -2x 4 + 4x 3 + 73. Write About It Explain the steps you would follow to write the polynomial 5x 2 - x - 3. 4x 3 - 3 + 5x 2 - 2x 4 - x in standard form. c 74. This problem will prepare you for the Multi-Step Test Prep on page 528. a. The perimeter of the rectangle shown is 12x + 6. What is the degree of this polynomial? 1 ÓÝÊÊÎ b. The area of the rectangle is 8x 2 + 12x. What is the degree of this polynomial? 2 7-6 PRACTICE A {Ý 7-6 PRACTICE C Practice B LESSON 7-6 Polynomials 7-6 PRACTICE B 500 Chapter 7 Exponents and Polynomials Find the degree and number of terms of each polynomial. 3 2. 7y 10y 2 1. 14h + 2h + 10 3 3 3. 2a 2 5a + 34 6a 4 2 2 4 4 LESSON 7-6 Write each polynomial in standard form. Then, give the leading coefficient. 4. 3x 2 2 + 4x 8 x 4x8 + 3x2 x 2 4 5. 7 50j + 3j 3 4j 2 3j3 4j 2 50j + 7 3 6. 6k + 5k 4 4k 3 + 3k 2 5k4 4k3 + 3k2 + 6k 5 Reading Strategies 7-6 READING STRATEGIES Understanding Vocabulary There is a great deal of introductory vocabulary related to polynomials. The meaning of and relationships among the terms are shown below. Classification Degree Constant 0 Linear 1 Quadratic 2 Cubic 3 Quartic 4 Quintic 5 6th degree 6 7th degree 7 etc. etc. A1NL11S_c07_0496-0501.indd 500 Classify each polynomial by its degree and number of terms. 7. 5t 2 + 10 8. 8w 32 + 9w 4 quadratic binomial quartic trinomial 9. b b 3 2b 2 + 5b 4 quartic polynomial 10. 3m + 8 2m 3 for m = 1 1 12. 2w + w w 2 for w = 2 2 3 7 9 10 b. How high is the egg above the ground after 6 seconds? 2. Why does 5xy 3 have a degree of 4? (Hint: Look at the exponents on the variables.) 3. Classify 4n 3 + 6 by degree. cubic by number of terms. binomial Use the polynomial 8g + 1 − 4g 2 to complete the following. 4. Write the polynomial in standard form. −4g 2 + 8g + 1 5. What is the leading coefficient? −4 6. What is the degree of the polynomial? 2 7. Classify the polynomial by number of terms. trinomial 500 8. Classify the polynomial by degree. quadratic Chapter 7 7-6 RETEACH The degree of the monomial is the sum of the exponents in the monomial. Find the degree of 8x 2 y 3 . Find the degree of −4a 6 b. 8x 2 y 3 The exponents are 2 and 3. −4a 6 b The exponents are 6 and 1. The degree of the monomial is 2 + 3 = 5. The degree of the monomial is 6 + 1 = 7. The degree of the polynomial is the degree of the term with the greatest degree. Find the degree of 2x 4 y 3 + 9x 5 . Find the degree of 4ab + 9a 3 . 2 x 4 y 3 + 9 x5 N 5 7 4 ab + 9 a3 N N 2 3 Degree of the polynomial is 7. Degree of the polynomial is 3. 5 x + 6 x3 + N 4 − 2 x4 N N N 4 1 3 0 Find the degree of each term. 2x 4 + 6x 3 + 5x + 4 Write the terms in order of degree. Find the degree of each monomial. Complete each of the following. The exponents on the variables have a sum of 4. 135.6 m Polynomials The leading coefficient is 2. 1. How many terms are in a trinomial? 3 a monomial? 1 187.5 m Review for Mastery Write 5x + 6x 3 + 4 + 2x 4 in standard form. 13. An egg is thrown off the top of a building. Its height in meters above the ground can be approximated by the polynomial 300 + 2t 4.9t 2, where t is the time since it was thrown in seconds. a. How high is the egg above the ground after 5 seconds? 7-6 A monomial is a number, a variable, or a product of numbers and variables with wholenumber exponents. A polynomial is a monomial or a sum or difference of monomials. The standard form of a polynomial is written with the terms in order from the greatest degree to the least degree. The coefficient of the first term is the leading coefficient. Evaluate each polynomial for the given value. 11. 4y 5 6y + 8y 2 1 for y = 1 LESSON 1. 7m 3 n 5 8 2. 6xyz 3. 4x 2 y 2 3 4 Find the degree of each polynomial. 4. x 5 + x 5 y 6 5. 4x 2 y 3 + y 4 + 7 5 6. x 2 + xy + y 2 Write each polynomial in standard form. Then give the leading coefficient. 7. x 3 − 5x 4 − 6x 5 −6x 5 − 5x 4 + x 3 −6 8. 2x + 5x 2 − x 3 −x 3 + 5x 2 + 2x −1 9. 8x + 7x 2 − 1 7x 2 + 8x −1 7 6/25/09 9:21:50 AM 75. In Exercise 76, students who chose A may have confused degree with number of terms. Students who chose D may think that the coefficients are significant when determining degree. /////ERROR ANALYSIS///// Two students evaluated 4x - 3x 5 for x = -2. Which is incorrect? Explain the error. 75. A is incorrect. The student incorrectly multiplied -3 by -2 before evaluating the power. -!+",!+" . 1/ . 1000/ 00/1 -!+",!+" . 1,!,+" 12/ 11 76. Which polynomial has the highest degree? 3x 8 - 2 x 7 + x 6 5x - 100 In Exercise 77, students who chose G probably got -3 for the value of the first term. 25x 10 + 3x 5 - 15 Answers 134x 2 77. What is the value of -3x 3 + 4x 2 - 5x + 7 when x = -1? 3 13 9 80b. Yes; 0 < x < 1; raising a number between 0 and 1 to a higher power results in a lesser number. So if x is between 0 and 1, the binomial with the least degree will have the greatest value. 19 78. Short Response A toy rocket is launched from the ground at 75 feet per second. The polynomial -16t 2 + 75t gives the rocket’s height in feet after t seconds. Make a table showing the rocket’s height after 1 second, 2 seconds, 3 seconds, and 4 seconds. At which of these times will the rocket be the highest? Journal CHALLENGE AND EXTEND 79. Medicine Doctors and nurses use growth charts and formulas to tell whether a baby is developing normally. The polynomial 0.016m 3 - 0.390m 2 + 4.562m + 50.310 gives the average length in centimeters of a baby boy between 0 and 10 months of age, where m is the baby’s age in months. a. What is the average length of a 2-month-old baby boy? a 5-month-old baby boy? 79c. The first 3 terms Round your answers to the nearest centimeter. 58 cm; 65 cm of the polynomial will equal 0, so just look at b. What is the average length of a newborn (0-month-old) baby boy? 50.310 cm c. How could you find the answer to part b without doing any calculations? the constant. 80. Consider the binomials 4x 5 + x, 4x 4 + x, and 4x 3 + x. a. Without calculating, which binomial has the greatest value for x = 5? 4x 5 + x b. Are there any values of x for 4x 3 + x which will have the greatest value? Explain. Have students explain how to classify a polynomial according to its degree and number of terms. Have students write a linear binomial and a quadratic trinomial, each with only x as a variable. Then have them evaluate each polynomial for x = 3. 7-6 SPIRAL REVIEW 81. Jordan is allowed 90 minutes of screen time per day. Today, he has already used m minutes. Write an expression for the remaining number of minutes Jordan has today. (Lesson 1-1) 90 - m 83. incons.; no sol. 84. cons. and dep.; inf. many solutions 82. Pens cost $0.50 each. Giselle bought p pens. Write an expression for the total cost of Giselle’s pens. (Lesson 1-1) 0.50p Classify each system. Give the number of solutions. (Lesson 6-4) ⎧ y = -4x + 5 83. ⎨ ⎩ 4x + y = 2 85. cons. and indep.; one sol. ⎧ 2x + 8y = 10 84. ⎨ ⎩ 4y = -x + 5 Simplify. (Lesson 7-4) 4 7 4 3, or 64 86. _ 44 _ x 6y 4 x 2 87. _ x 4y 9 y 5 ( ) 2v 4 _ vw 5 2 4v _ ( ) 89. w 10 _ 7- 6 Polynomials Name ________________________________________ Date __________________ Class__________________ LESSON 7-6 Problem Solving 7-6 PROBLEM SOLVING Polynomials Write the correct answer. 1. The surface area of a cylinder is given by the polynomial 2πr 2 + 2πrh. A cylinder has a radius of 2 centimeters and a height of 5 centimeters. Find the surface area of the cylinder. Use 3.14 for π. A1NL11S_c07_0496-0501.indd 501 2. A firework is launched from the ground at a velocity of 180 feet per second. Its height after t seconds is given by the polynomial −16t 2 + 180t. Find the height of the firework after 2 seconds and after 5 seconds. 87.92 square centimeters 2 s: 296 feet 5 s: 500 feet 3. In the United Kingdom, transportation 1 2 v +v authorities use the polynomial 20 4. A piece of cardboard that measures 2 feet by 3 feet can be folded into a box if notches are cut out of the corners. The length of the side of the notch will be the same as the height h of the resulting box. The volume of the box is given by 4h 3 − 10 h 2 + 6h. Find the volume of the box for h = 0.25 and h = 0.5. for calculating the number of feet needed to stop on dry pavement. In the United States, many use the polynomial 0.096v 2 . Both formulas are based on speed v in miles per hour. Calculate the stopping distances for a car traveling 45 miles per hour in both the U.S. and the UK. h = 0.25: 0.9375 cubic feet h = 0.5: 1 cubic foot UK: 146.25 feet US: 194.4 feet The height of a rocket in meters t seconds after it is launched is approximated by the polynomial 0.5at 2 + vt + h where a is always −9.8, v is the initial velocity, and h is the initial height. Use this information with the data in the chart for questions 5 – 7. Select the best answer. 5. A 300X was launched from a height of 10 meters. What was its height after 3 seconds? A 715.9 m B 745.3 m Model Number 300X Q99 4400i C 755.5 m D 760 m 6. Marie and Bob launched their rockets at the same time from a platform 5 meters above the ground. Marie launched the 4400i and Bob launched the Q99. How much higher was Marie’s rocket after 2 seconds? 7-6 Challenge 7-6 CHALLENGE Pick the Polynomial Match each polynomial with the correct clue. Each polynomial can be used only once. Not every polynomial will be used. Use these polynomials for 1 – 7. 5 3 x +x +x 2x 4 3x + 3y + 3z 2xy + 5xz 4x 2 3x 5 + x 4 4 Use these polynomials for 8 – 14. 2 5 2 1. I am a monomial with degree 5. Who am I? 7. The 4400i was launched from the ground at the same time the Q99 was launched from 175 meters above the ground. After how many seconds were the rockets at the same height? H 140 meters A 2s C5 s G J 320 meters B 4s D 6s 3 2 x +x +x x4 + x3 + x2 2x 2 3x 3 + 1 x3 x2 + y2 + z2 + w2 8. I am a linear expression. My constant is 3. Who am I? 3xy4 x 3 2. I am a sum of monomials with degree 8. Who am I? 9. I am a quartic trinomial. I have three different variables. Who am I? x3 y 3xyz + z 2x4 y4 + 3x 2 y5 3. I am a trinomial with degree 5. Who am I? 10. I am a cubic binomial. Who am I? 4x2 3x 2 y x5 + x3 + x 11. I am a quadratic polynomial. I have no constants. Who am I? x2 + y2 + z 2 + w2 2xy + 5xz Initial Velocity (m/s) 250 90 125 2 4x 3x y x2 + x 3 3 + x + 4x 2 x 3 y 3xyz + z x + 3 2x y + 3x y x 5 y 3x 2 y xy 3xy 4 2x 2 y + 5xy 2 4xyz 2 + xyz 4. I am a binomial. Both of my terms have degree 2. Who am I? F 35 meters 70 meters Name _______________________________________ Date __________________ Class__________________ LESSON 12. I am a cubic trinomial with one variable. Who am I? 5. I am a monomial with degree 4. Who am I? x3 + x2 + x 2x4 13. I am a quadratic trinomial. When you put me in standard form, my leading coefficient is 4. Who am I? 6. I am a binomial with degree 4. Who am I? 4xyz 2 + xyz 3 + x + 4x2 7. I am a trinomial. When you put me in standard form, my leading coefficient is 3. Who am I? 2. 25x2 - 3x4 14. I am a quartic trinomial. Who am I? x4 + x3 + x2 7-49 Holt McDougal Algebra 1 7 4. 14 - x4 + 3x2 -x4 + 3x2 + 14; -1 501 Classify each polynomial according to its degree and number of terms. 5. 18x2 - 12x + 5 quadratic trinomial 6/25/09 9:21:57 AM 6. 2x4 - 1 quartic binomial 7. The polynomial 3.675v + 0.096v2 is used to estimate the stopping distance in feet for a car whose speed is v miles per hour on flat, dry pavement. What is the stopping distance for a car traveling at 70 miles per hour? 727.65 ft Also available on transparency 4x2 3x5 + x Lesson 7-6 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 5 4 3. 24g3 + 10 + 7g5 - g2 7g5 + 24g3 - g2 + 10; 2p -4 p 8 _ p3 16 6 1. 7a3b2 - 2a4 + 4b - 15 Write each polynomial in standard form. Then give the leading coefficient. ⎧ y = 3x + 2 85. ⎨ ⎩ y = -5x - 6 88. Find the degree of each polynomial. Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-48 Holt McDougal Algebra 1 501 7-7 Organizer Model Polynomial Addition and Subtraction Use with Lesson 7-7 Pacing: 1 Traditional __ day 2 1 __ Block 4 day You can use algebra tiles to model polynomial addition and subtraction. Objective: Use algebra tiles to Use with Lesson 7-7 model polynomial addition and subtraction. KEY GI Materials: algebra tiles @<I <D =1 = -1 Online Edition KEYWORD: MA11 LAB7 =x = x2 = -x = -x 2 Algebra Tiles Countdown Week 16 Activity 1 Use algebra tiles to find (2x 2 - x) + (x 2 + 3x - 1). Resources MODEL Algebra Lab Activities ALGEBRA 7-7 Lab Recording Sheet Teach Use tiles to represent all terms from both expressions. (2x 2 - x) + (x 2 + 3x - 1) Rearrange tiles so that like tiles are together. Like tiles are the same size and shape. (2x 2 + x 2) + (-x + 3x) - 1 Discuss Compare like terms and like tiles. In like terms, the same variables are raised to the same powers, but the coefficients may differ. In like tiles, the size and shape are the same, but the color of the tile (whether it is positive or negative) may differ. Remove any zero pairs. In Activity 2, remind students to add the opposite of every term in the second expression. The remaining tiles represent the sum. Encourage students to always arrange their tiles and write their answers in standard form. This will help prevent careless errors. 3x 2 - x + x + 2x - 1 3x 2 + 2x - 1 Try This Use algebra tiles to find each sum. 2. (3x 2 + 2x + 5) + (x 2 - x - 4) 4x 2 + x + 1 3. (x - 3) + (2x - 2) 3x - 5 4. (5x 2 - 3x - 6) + (x 2 + 3x + 6) 6x 2 5. -5x 2 + (2x 2 + 5x) -3x 2 + 5x 6. (x 2 - x - 1) + (6x - 3) x 2 + 5x - 4 1. 502 (-2x 2 + 1) + (-x 2) -3x 2 + 1 Chapter 7 Exponents and Polynomials A11NLS_c07_0502-0503.indd 502 KEYWORD: MA7 Resources 502 Chapter 7 12/10/09 8:48:46 PM Close Activity 2 Key Concept Use algebra tiles to find (2x 2 + 6) - 4x 2. MODEL Polynomial addition is the same as combining like terms. Polynomial subtraction can be written as polynomial addition by adding the opposite. ALGEBRA Use tiles to represent the terms in the first expression. 2x 2 + 6 Assessment Journal Have students explain how to use algebra tiles to find (3x2 - 4x + 1) - 5x. To subtract 4x 2, you would remove 4 yellow x 2-tiles, but there are not enough to do this. Remember that subtraction is the same as adding the opposite, so rewrite (2x 2 + 6) - 4x 2 as (2x 2 + 6) + (-4x 2). Answers to Try This 7. 3x2 + 4x 8. 2x2 - 4x - 7 2x 2 + 6 + (-4x 2) Add 4 red x 2-tiles. 9. 3x 10. 10x + 5 11. 5x2 + x 12. 2x2 - 6x - 4 Rearrange tiles so that like tiles are together. 2x + (-4x )+ 6 2 13. 2 2x 2 + (-2x 2) + (-2x 2) + 6 Remove zero pairs. The remaining tiles represent the difference. -2x 2 + 6 Try This Use algebra tiles to find each difference. (6x 2 + 4x) - 3x 2 8. 10. (8x + 5) - (-2x) 11. 7. 13. (2x 2 + x - 7) - 5x (x 2 + 2x) - (-4x 2 + x) 9. (3x + 6) - 6 12. (3x 2 - 4) - (x 2 + 6x) represents a zero pair. Use algebra tiles to model two other zero pairs. 14. When is it not necessary to “add the opposite” for polynomial subtraction using algebra tiles? when you have enough tiles to actually remove them to model the subtraction 7- 7 Algebra Lab A1NL11S_c07_0502-0503.indd 503 503 7/18/09 5:01:25 PM 7-7 Algebra Lab 503 7-7 Organizer 7-7 Pacing: Traditional 1 day Block __1 day Adding and Subtracting Polynomials 2 Objective: Add and subtract Who uses this? Business owners can add and subtract polynomials that model profit. (See Example 4.) Objective Add and subtract polynomials. Technology Lab GI In Technology Lab Activities <D @<I Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms. Online Edition Tutorial Videos Countdown Week 16 EXAMPLE 1 www.cartoonstock.com polynomials. Adding and Subtracting Monomials Add or subtract. A 15m 3 + 6m 2 + 2m 3 Warm Up 15m 3 + 6m 2 + 2m 3 15m 3 + 2m 3 + 6m 2 3 17m + 6m 2 Simplify each expression by combining like terms. 1. 4x + 2x 6x 2. 3y + 7y 10y 3. 8p - 5p 3p 4. 5n + 6n2 not like terms Simplify each expression. 5. 3(x + 4) Identify like terms. Rearrange terms so that like terms are together. Combine like terms. B 3x 2 + 5 - 7x 2 + 12 Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see Lesson 1-7. 3x 2 + 5 - 7x 2 + 12 2 3x - 7x 2 + 5 + 12 -4x 2 + 17 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. C 0.9y - 0.4y + 0.5x + y 5 5 5 5 0.9y 5 - 0.4y 5 + 0.5x 5 + y 5 0.9y 5 - 0.4y 5 + y 5 + 0.5x 5 1.5y 5 + 0.5x 5 3x + 12 6. -2(t + 3) -2t - 6 Identify like terms. Rearrange terms so that like terms are together. Combine like terms. D 2x y - x y - x y 2 2 2 2x 2y - x 2y - x 2y 0 7. -1( - 4x - 6) -x2 + 4x + 6 x2 Also available on transparency All terms are like terms. Combine. Add or subtract. 1a. 2s 2 + 3s 2 + s 5s 2 + s 1b. 4z 4 - 8 + 16z 4 + 2 20z 4 - 6 8 8 8 8 8 8 1c. 2x + 7y - x - y x + 6y 1d. 9b 3c 2 + 5b 3c 2 - 13b 3c 2 b 3c 2 Polynomials can be added in either vertical or horizontal form. In vertical form, align the like terms and add: Teacher: What is b + b? Shakespeare: Is it 2b or not 2b? (5x 2 + 4x + 1) + (2x 2 + 5x + 2) = (5x 2 + 2x 2) + (4x + 5x) + (1 + 2) 5x 2 + 4x + 1 + 2x 2 + 5x + 2 7x 2 + 9x + 3 504 In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms: = 7x 2 + 9x + 3 Chapter 7 Exponents and Polynomials 1 Introduce A1NL11S_c07_0504-0509.indd e x p l o r504 at i o n 7-7 Adding and Subtracting Polynomials An ecologist is studying frogs and toads in a wetlands habitat. She ﬁnds that the polynomial 3x 2 ⫹ x models the frog population and that the polynomial 7x 2 ⫺ x models the toad population. In both cases, x represents the number of months. 1. Complete the table. For each month, evaluate the two polynomials to find the population of frogs and toads. Then add these values to find the total population. Population of Population of Month, x Frogs, 3x 2 ⫹ x Toads, 7x 2 ⫺ x 1 4 6 2 3 4 5 6 Total Population 10 2. Look for a pattern in the last column. Write a simplified polynomial that gives the total population in month x. KEYWORD: MA7 Resources THINK AND DISCUSS 3. Show how to find the sum of the polynomials 3x 2 ⫹ x and 7x 2 ⫺ x. 4. Describe how you could find the sum using the terms of the two polynomials that are being added. 504 Chapter 7 Motivate Display the following items: 4 books, 3 pencils, 2 books, and 5 pencils. Ask students for a sensible way to group the items. a group of 6 books and a group of 8 pencils Explain to students that adding and subtracting monomials is done in a similar way. Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 6/25/09 9:19:47 AM EXAMPLE 2 Adding Polynomials "" A (2x 2 - x) + (x 2 + 3x - 1) (2x 2 - x) + (x 2 + 3x - 1) (2x 2 + x 2) + (-x + 3x) + (-1) 3x 2 + 2x - 1 Students often think the terms xy2 and x2y are like terms. Write out all factors to show students how they are different: xy2 = xyy, and x2y = xxy. Identify like terms. Group like terms together. Combine like terms. B (-2ab + b) + (2ab + a) When you use the Associative and Commutative Properties to rearrange the terms, the sign in front of each term must stay with that term. C (-2ab + b) + (2ab + a) (-2ab + 2ab) + b + a Identify like terms. 0+b+a b+a Combine like terms. Group like terms together. (4b + 8b) + (3b + 6b - 7b + b) (4b 5 + 8b) + (3b 5 + 6b - 7b 5 + b) (4b 5 + 8b) + (-4b 5 + 7b) 5 5 4b + 8b + -4b 5 + 7b 5 + 15b 0 15b D Additional Examples Simplify. 5 Example 1 Identify like terms. Add or subtract. Combine like terms in the second polynomial. Use the vertical method. A. 12p3 + 11p2 + 8p3 20p3 + 11p2 Combine like terms. B. 5x2 - 6 - 3x + 8 5x2 - 3x + 2 Simplify. (20.2y + 6y + 5) + (1.7y - 8) (20.2y 2 + 6y + 5) + (1.7y 2 - 8) 2 C. t2 + 2s2 - 4t2 - s2 -3t 2 + s2 2 20.2y 2 + 6y + 5 + 1.7y 2 + 0y - 8 21.9y 2 + 6y - 3 D. 10m2n + 4m2n - 8m2n 6m2n Identify like terms. Use the vertical method. Example 2 Write 0y as a placeholder in the second polynomial. Add. A. (4m2 + 5) + (m2 - m + 6) 5m2 - m + 11 Combine like terms. 2. Add (5a 3 + 3a 2 - 6a + 12a 2) + (7a 3 - 10a). 12a 3 + 15a 2 - 16a To subtract polynomials, remember that subtracting is the same as adding the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial: - (2x 3 - 3x + 7) = -2x 3 + 3x - 7 EXAMPLE 3 Ê,,", ,/ Add. Subtracting Polynomials B. (10xy + x) + (-3xy + y) 7xy + x + y C. (6x2 - 4y) + (3x2 + 3y 8x2 - 2y) x2 - 3y 1 D. _ a2 + b + 2 + 2 3 2 _ a - 4b + 5 2a2 - 3b + 7 2 ( ( ) ) Subtract. A (2x 2 + 6) - (4x 2) (2x 2 + 6) + (- 4x 2) (2x 2 + 6) + (-4x 2) (2x 2 - 4x 2) + 6 -2x 2 + 6 Rewrite subtraction as addition of the opposite. Identify like terms. Questioning Strategies Group like terms together. Combine like terms. EX AM P LE B (a 4 - 2a) - (3a 4 - 3a + 1) (a 4 - 2a) + (- 3a 4 + 3a - 1) Rewrite subtraction as addition of the opposite. (a 4 - 2a) + (- 3a 4 + 3a - 1) Identify like terms. (a 4 - 3a 4) + (- 2a + 3a) - 1 Group like terms together. -2a 4 + a - 1 Combine like terms. 7- 7 Adding and Subtracting Polynomials 6/25/09 Guided Instruction Review like terms before beginning this lesson. When rearranging terms, remind students to pay close attention to the sign in front of each term. Point out that in Example 2C, like terms in the second polynomial were simplified before the polynomials were added. Students will be less likely to make mistakes when the individual polynomials are simplified first. Visual Cues Draw different marks around like terms so they stand out. Through Cooperative Learning Prepare a bag with several small slips of paper (at least four times the number of students in class), each with one of the following: x, 2x, -3x, y, 4y, -6y. Have each student randomly select three or four slips of paper. Have students combine, and simplify if possible, their terms and write the resulting polynomial on a piece of paper. Then have students pair up to find the sums and differences of their polynomials. Have students pair up with as many others as time allows. 1 • How do you identify like terms? EX AM P LE 2 • When using a vertical format to add polynomials, what is a placeholder? How is it helpful? 505 2 Teach A1NL11S_c07_0504-0509.indd 505 INTERVENTION Inclusion Remind students that the Commutative Property of Addition states that you can 9:19:56 AM numbers in any order. The add Associative Property of Addition states that you can group any of the numbers together. Lesson 7-7 505 Multiple Representations Finding the opposite of a polynomial can be thought of as distributing -1. Subtract. C (3x 2 - 2x + 8) - (x 2 - 4) (3x 2 - 2x + 8) + (- x 2 + 4) (3x 2 - 2x + 8) + (- x 2 + 4) 3x 2 - 2x + 8 + -x 2 + 0x + 4 Additional Examples 2x 2 - 2x + 12 (11z 3 - 2z) + (- z 3 + 5) (11z 3 - 2z) + (- z 3 + 5) Subtract. -x3 + 4y 11z 3 - 2z + 0 + -z 3 + 0z + 5 B. (7m4 - 2m2) (5m4 - 5m2 + 8) 2m4 + 3m2 - 8 10z - 2z + 5 3 C. (-10x2 - 3x + 7) - (x2 - 9) -11x2 - 3x + 16 EXAMPLE 4 A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of plot A can be represented by 3x2 + 7x - 5, and the area of plot B can be represented by 5x2 - 4x + 11. Write a polynomial that represents the total area of both plots of land. 8x2 + 3x + 6 (- 0.03x 2 + 25x - 1500) - (-0.02x 2 + 21x - 1700) + (+ 0.02x 2 - 21x + 1700) Write 0 and 0z as placeholders. Combine like terms. Southern: 0.02x 2 21x 1700 Eastern: 0.03x2 25x 1500 Eastern plant profits Southern plant profits Write subtraction as addition of the opposite. Combine like terms. -0.05x 2 + 46x - 3200 3 THINK AND DISCUSS • What is the first step in subtracting polynomials? 1. Identify the like terms in the following list: -12x 2, -4.7y, __15 x 2y, y, 3xy 2, -9x 2, 5x 2y, -12x • Why are parentheses used when subtracting polynomials? 2. Describe how to find the opposite of 9t 2 - 5t + 8. 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write an example that shows how to perform the given operation. 4 • What is the purpose of aligning terms before adding or subtracting polynomials? 506 *Þ>Ã ``} -ÕLÌÀ>VÌ} Chapter 7 Exponents and Polynomials Answers to Think and Discuss 3 Close A1NL11S_c07_0504-0509.indd 506 Have students list the steps for adding and subtracting polynomials. 1. Rewrite subtraction as addition if necessary. 2. Identify like terms. 3. Rearrange terms so that like terms are together. Chapter 7 Use the vertical method. 4. Use the information above to write a polynomial that represents the total profits from both plants. Questioning Strategies 506 Identify like terms. -0.03x + 25x - 1500 INTERVENTION 5. Simplify if necessary. Rewrite subtraction as addition of the opposite. 2 -0.01x + 4x + 200 4. Combine like terms. Combine like terms. Write a polynomial that represents the difference of the profits at the eastern plant and the profits at the southern plant. 2 Summarize Write 0x as a placeholder. Business Application The profits of two different manufacturing plants can be modeled as shown, where x is the number of units produced at each plant. Example 4 EX A M P L E Use the vertical method. 3. Subtract (2x 2 - 3x 2 + 1) - (x 2 + x + 1). -2x 2 - x D. (9q2 - 3q) - (q2 - 5) 8q2 - 3q + 5 EX A M P L E Identify like terms. D (11z 3 - 2z) - (z 3 - 5) Example 3 A. (x3 + 4y) - (2x3) Rewrite subtraction as addition of the opposite. and INTERVENTION Diagnose Before the Lesson 7-7 Warm Up, TE p. 504 1. -12x2 and -9x2; -4.7y and 1 y; _ x2y and 5x2y 5 2. Take the opposite of each term: -9t2 + 5t - 8. 3. See p. A7. Monitor During the Lesson Check It Out! Exercises, SE pp. 504–506 Questioning Strategies, TE pp. 505–506 Assess After the Lesson 7-7 Lesson Quiz, TE p. 509 Alternative Assessment, TE p. 509 6/25/09 9:19:59 AM 7-7 Exercises 7-7 Exercises KEYWORD: MA11 7-7 KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 Assignment Guide Add or subtract. p. 504 1. 7a 2 - 10a 2 + 9a 2. 13x 2 + 9y 2 - 6x 2 3. 0.07r 4 + 0.32r 3 + 0.19r 4 _ 5. 5b 3c + b 3c - 3b 3c 6. -8m + 5 - 16 + 11m -3a 2 + 9a 7x + 9y 2 1 p3 + _ 2 p3 4. _ 11 p 3 4 3 2 Add. p. 505 7. 9. SEE EXAMPLE 3 p. 505 (5n 3 + 3n + 6) + (18n 3 + 9) (-3x + 12) + (9x 2 + 2x - 18) 8. 10. (6c 4 + 8c + 6) - (2c 4) 4c 4 + 8c + 6 12. 13. (2r + 5) - (5r - 6) -3r + 11 SEE EXAMPLE 4 p. 506 3 3m - 11 (3.7q 2 - 8q + 3.7) + (4.3q 2 - 2.9q + 1.6) (9x 4 + x 3) + (2x 4 + 6x 3 - 8x 4 + x 3) 10y 2 - 13y + 9 Subtract. 11. 4 3b 3c 12 SEE EXAMPLE 0.26r + 0.32r 2 14. (16y 2 - 8y + 9) - (6y 2 - 2y + 7y) (-7k 2 + 3) - (2k 2 + 5k - 1) -9k 2 - 5k + 4 15. Geometry Write a polynomial that represents the measure of angle ABD. nÊÊÊÊÊÊÓ>ÊÊx® Ê>ÊÓÊ Â 8a + 5a + 9 2 16–24 25–28 29–32 33–34 1 2 3 4 Quickly check key concepts. Exercises: 24, 28, 30, 32, 33, 48 Add or subtract. 16. 4k 3 + 6k 2 + 9k 3 17. 5m + 12n 2 + 6n - 8m 18. 2.5a 4 - 8.1b 4 - 3.6b 4 19. 2d + 1 - d 20. 7xy - 4x y - 2xy 21. -6x 3 + 5x + 2x 3 + 4x 3 23. 3x 3 - 4 - x 3 - 1 24. 3b 3 - 2b - 1 - b 3 - b 5 5 22. x 2 + x + 3x + 2x 2 2 Add. Extra Practice Skills Practice p. S17 Application Practice p. S34 25. 27. (2t 2 - 8t) + (8t 2 + 9t) 10t 2 + t (x 5 - x) + (x 4 + x) x 5 + x 4 26. 28. If you finished Examples 1–4 Basic 16–43, 45–51, 53–56, 63–72 Average 16–56, 58, 63–72 Advanced 16–44, 52–72 Homework Quick Check PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example If you finished Examples 1–2 Basic 16–28 Average 16–28, 58 Advanced 16–28, 58 Ç>ÊÊ{®Â Assign Guided Practice exercises as necessary. Auditory For Exercises 1–6, suggest to students that they say each term out loud as they do their homework to hear the difference between like and unlike terms. -3x 2 - 11x + 3 (-7x 2 - 2x + 3) + (4x 2 - 9x) (-2z 3 + z + 2z 3 + z) + (3z 3 - 5z 2) 3z 3 - 5z 2 + 2z Subtract. 29. 31. (t 3 + 8t 2) - (3t 3) -2t 3 + 8t 2 (5m + 3) - (6m 3 - 2m 2) 30. -6m 3 + 2m 2 + 5m + 3 32. (3x 2 - x) - (x 2 + 3x - x) 2x 2 - 3x (3s 2 + 4s) - (-10s 2 + 6s) 13s 2 - 2s 33. Photography The measurements of a photo and its frame are shown in the diagram. Write a polynomial that represents the width of the photo. 4w 2 + 6w + 4 34. Geometry The length of a rectangle is represented by 4a + 3b, and its width is represented by 7a - 2b. Write a polynomial for the perimeter of the rectangle. 22a + 2b ÈÜ ÓÊÊn ¶ Ü ÓÊÎÜÊÊÓ 7- 7 Adding and Subtracting Polynomials Answers 507 24. 2b3 - 3b - 1 7. 23n3 + 3n + 15 8. 8q2 - 10.9q + 5.3 A1NL11S_c07_0504-0509.indd 507 9. 9x2 -x-6 10. 3x4 + 8x3 16. 13k3 6/25/09 6:10:10 PM + 6k2 17. 12n2 + 6n - 3m 18. 2.5a4 - 11.7b4 19. d5 + 1 20. -4x2y + 5xy 21. 5x 22. 3x2 + 4x 23. 2x3 - 5 KEYWORD: MA7 Resources Lesson 7-7 507 Exercise 53 involves writing expressions for the dimensions of a rectangle. This exercise prepares students for the Multi-Step Test Prep on page 528. 41. -u + 3u + 3u + 6 Answers 43. Add or subtract. 35. (2t - 7) + (-t + 2) t - 5 37. (4n - 2) - 2n 2n - 2 3 52. No; polynomial addition simply involves combining like terms. No matter what order the terms are combined in, the sum will be the same. Yes; in polynomial subtraction, the subtraction sign is distributed among all terms in the second polynomial, changing all the signs to their opposites. 38. (-v - 7) - (-2v) v - 7 6x 2 - x - 1 (4x + 3x - 6) + (2x - 4x + 5) (5u 2 + 3u + 7) - (u 3 + 2u 2 + 1) 2 39. 41. 2 (4m 2 + 3m) + (-2m 2) 2m 2 + 3m 4z 2 - 10z - 4 40. (2z 2 - 3z - 3) + (2z 2 - 7z - 1) 42. (-7h 2 - 4h + 7) - (7h 2 - 4h + 11) -14h 2 - 4 43. Geometry The length of a rectangle is represented by 2x + 3, and its width is represented by 3x + 7. The perimeter of the rectangle is 35 units. Find the value of x. _3 , or 1.5 2 44. Write About It If the parentheses are removed from (3m 2 - 5m) + 2 equivalent to the original? If the 44. Yes; the simplified (12m + 7m - 10), is the new expression parentheses are removed from (3m 2 - 5m) - (12m 2 + 7m - 10), is the new form of both expressions expression equivalent to the original? Explain. is 15m 2 + 2m - 10. No; the simplified 45. /////ERROR ANALYSIS///// Two students found the sum of the polynomials form of the orig. (-3n 4 + 6n 3 + 4n 2) and (8n 4 - 3n 2 + 9n). Which is incorrect? Explain expression is the error. -9m 2 - 12m + 10, and the simplified form ! " of the new expression is ,g -/g ,-g +)g ,g -/g ,-g + -9m 2 + 2m - 10. -)g ,,g +2g 1g -,g +2g 1 g XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 45. B is incorrect. The student .g -/g ,g +2g .g -,g ,*,g + incorrectly tried to combine 6n 3 and -3n 2, which are not like terms, and 4n 2 and Copy and complete the table by finding the missing polynomials. 9n, which are not like terms. x 53a. 2 36. x Polynomial 1 Polynomial 2 Sum 46. x -6 3x - 10x + 2 4x - 10x - 4 47. 12x + 5 3x + 6 15x + 11 48. x 4 - 3x 2 - 9 5x 4 + 8 6x 4 - 3x 2 - 1 49. 7x 3 - 6x - 3 6x + 14 7x 3 + 11 50. 2x + 5x 7x - 5x + 1 9x 3 + 1 x+x +6 3x 2 + 2x + 1 2 3 2 2 2x 2 + x - 5 51. 3 2 2 2 52. Critical Thinking Does the order in which you add polynomials affect the sum? Does the order in which you subtract polynomials affect the difference? Explain. 53. This problem will prepare you for the Multi-Step Test Prep on page 528. a. Ian plans to build a fenced dog pen. At first, he planned for the pen to be a square of length x feet on each side, but then he decided that a square may not be best. He added 4 feet to the length and subtracted 3 feet from the width. Draw a diagram to show the dimensions of the new pen. b. Write a polynomial that represents the amount of fencing that Ian will need for the new dog pen. 4x + 2 c. How much fencing will Ian need if x = 15? 62 ft 7-7 PRACTICE A 7-7 PRACTICE C Practice B LESSON 7-7 7-7 PRACTICE B 508 Adding and Subtracting Polynomials Chapter 7 Exponents and Polynomials Add or subtract. 12m 3 2m 2 3 1. 3m3 + 8m3 3 + m3 2m2 –7p 5 10pg + 5g 2. 2pg p5 12pg + 5g 6p5 Add. 3. 3k2 2k + 7 + 5x2 2x + 3y 4. k2 3k2 k + 5 2 5. 11hz3 + 3hz2 + 8hz + 6x2 + 5x + 6y + 9hz3 + hz2 3hz 11x2 + 3x + 9y 20hz 3 + 4hz 2 + 5hz 2 6. (ab + 13b 4a) + (3ab + a + 7b) 2 3 7-7 3 5x 2x 2 7. (4x x + 4x) + (x x 4x) 12d2 + 3dx + x 2v5 3v4 8 9. (4d2 + 2dx 8x) (3v5 + 2v4 8) 16d 2 + dx + 9x a v5 5v4 11. (r2 + 8pr p) (12r2 2pr + 8p) 2 3 3 2 12. (un n + 2un ) (3un + n + 4un) 10. Subtract (x2 + 8x − 4) − (3x2 − 3x + 2). 7 − 3(x + 8) + 4x (x2 + 8x − 4) − (3x2 − 3x + 2) 2 33b 8 14. Darnell and Stephanie have competing refreshment stand businesses. Darnell’s profit can be modeled with the polynomial c2 + 8c 100, where c is the number of items sold. Stephanie’s profit can be modeled with the polynomial 2c2 7c 200. a. Write a polynomial that represents the difference between Stephanie’s profit and Darnell’s profit. c2 15c 100 b. Write a polynomial to show how much they can expect to earn if they decided to combine their businesses. 3c + c 300 3 = x − 17 = x2 − 3x2 + 8x + 3x − 4 − 2 = −2x2 + 11x − 6 1. What is being distributed in the linear expression on the left? −3 2. What is being distributed in the polynomial subtraction on the right? −1 The following are not like terms: 6/25/09 9:20:14 AM −3x and 4x; 7 and −24 x 2 and −3x 2 ; 8x and 3x; −4 and −2 2x 3 + 2x + 7 9x 2 + 16x + 2 Rearrange terms so that like terms are together. 8x2 + 10x Combine like terms. Add (5y2 + 7y + 2) + (4y2 + y + 8). (5y2 + 7y + 2 ) + (4y2 + y + 8 ) Identify like terms. (5y2 + 4y2) + ( 7y + y ) + ( 2 + 8 ) Rearrange terms so that like terms are together. 9y2 + 8y + 10 Combine like terms. no; same variable raised to different power 1. 4x and x4 3 3. 2z and 4x yes; same variable raised to same power 3 no; different variable raised to same power Add. 4. 2y2 + 3y + 7y + y2 Add or subtract the polynomials. 7. (2x2 + 10x + 4) + (7x2 + 6x − 2) Identify like terms. 3x2 + 5x2 + 4x + 6x 2. 5y and 7y 4. Identify the sets of like terms that were combined in the polynomial subtraction on the right. 5. 5x3 + 2x + 1 − 3x3 + 6 Add 3x2 + 4x + 5x2 + 6x. 3x2 + 4x + 5x2 + 6x Determine whether the following are like terms. Explain. 3. Identify the sets of like terms that were combined in the expression on the left. 4x 4 + 2x − 2 Chapter 7 Adding and Subtracting Polynomials The following are like terms: = x + 8x − 4 − 3x + 3x − 2 Complete the following based on the examples above. 9. (6x4 + 8x − 2) − (2x4 + 6x) 508 7-7 RETEACH 7-7 You can add or subtract polynomials by combining like terms. 2 Step 3: Combine all the sets of like terms. 11r 2 + 10pr 9p 13. Antoine is making a banner in the shape of a triangle. He wants to line the banner with a decorative border. How long will the border be? 2 = −3x + 4x + 7 − 24 5y4 + 8ay2 2y + 3un 2n un 2 Step 2: Rearrange so like terms are together. y4 + 6ay2 y + a Review for Mastery LESSON Step 1: Use the Distributive Property. = 7 − 3x − 24 + 4x (6y4 2ay 2 + y) 2 7-7 READING STRATEGIES Connecting Concepts Simplify 7 − 3(x + 8) + 4x. Subtract. 8. Reading Strategies 4ab + 20b 3a A1NL11S_c07_0504-0509.indd 508 2 3 LESSON The process for adding and subtracting polynomials is the same as the process for simplifying linear expressions. Look at the connections below. 6. x − 3x5 + 2x4 − 5x5 − x −8x 5 + 2x 4 8. (x3 − 6) + (9 − 2x2 + x3) 2x 3 − 2x 2 + 3 10. (3x2 − 9x) − (x + 2x3 − 4) −2x 3 + 3x 2 − 10x + 4 2 3y + 10y 5. 8m4 + 3m 4m4 4 4m + 3m 7. (6x2 + 3x) + (2x2 + 6x) 8. (m2 10m + 5) + (8m + 2) 9. (6x3 + 5x) + (4x3 + x2 2x + 9) 10. (2y5 6y3 + 1) + (y5 + 8y4 2y3 1) 6. 12x5 + 10x4 + 8x4 12x5 + 18x4 8x2 + 9x m2 2m + 7 10x3 + x2 + 3x + 9 3y5 + 8y4 8y 3 In Exercise 54, after determining that the missing term is a y-term, encourage students to write and solve an algebraic equation to find the coefficient: -12 + x - 6 = -15. 54. What is the missing term? (-14y 2 + 9y 2 - 12y + 3) + (2y 2 + -6y - 6y - 2) = (-3y 2 - 15y + 1) -3y 3y 6y 55. Which is NOT equivalent to -5t - t? 3 -(5t 3 + t) (t 3 + 6t) - (6t 3 + 7t) (2t 3 - 3t 2 + t) - (7t 3 - 3t 2 + 2t) (2t 3 - 4t) - (-7t - 3t) 56. Extended Response Tammy plans to put a wallpaper border around the perimeter of her room. She will not put the border across the doorway, which is 3 feet wide. 56b. 7; If x = 7, a. Write a polynomial that represents the Tammy will need number of feet of wallpaper border that Tammy will need. 6x + 3 6(7) + 3 = 45 feet of wallpaper border. b. A local store has 50 feet of the border that However, if x = 8, Tammy has chosen. What is the greatest Tammy will need whole-number value of x for which this 6(8) + 3 = 51 feet amount would be enough for Tammy’s room? Justify your answer. of wallpaper border, which is more than c. Determine the dimensions of Tammy’s room for the store has. the value of x that you found in part b. 13 ft × 11 ft Answers 63. À 64. ÝÊÊ{®ÊvÌ 58. 2m 3 + 2m, 2m 3 + m ÓÝÊÊ£®ÊvÌ Journal Have students describe two different ways of adding polynomials, using examples in their descriptions. 57. Geometry The legs of the isosceles triangle at right measure (x 3 + 5) units. The perimeter of the triangle is (2x 3 + 3x 2 + 8) units. Write a polynomial that represents the measure of the base of the triangle. 3x 2 - 2 ÊÝ ÊÎ Êx ÊÝ ÊÎ Êx Have students write four different polynomials, using any of the following as the variable part of the terms: x, y, x2, y2. Instruct students to pick any two of the polynomials and find the sum. Then have them find the difference of the remaining two polynomials. 59. Write two polynomials whose difference is 4m 3 + 3m. 60. Write three polynomials whose sum is 4m 3 + 3m. 60. 2m 3 + m, m 3 + m, m 3 + m 66–68. For graphs, see p. A27. 58. Write two polynomials whose sum is 4m 3 + 3m. 59. 5m 3 + 2m, m3 - m 65. CHALLENGE AND EXTEND 58–62. Possible answers given. 61. Write two monomials whose sum is 4m 3 + 3m. 4m 3, 3m 62. Write three trinomials whose sum is 4m 3 + 3m. 2m 3 + m 2 + m, m 3 + m 2 + m, m 3 - 2m 2 + m SPIRAL REVIEW Solve each inequality and graph the solutions. (Lesson 3-2) 63. d + 5 ≥ -2 d ≥ -7 64. 15 < m - 11 m > 26 65. -6 + t < -6 t < 0 7-7 Write each equation in slope-intercept form. Then graph the line described by each equation. (Lesson 5-7) 1 x + 6 y = 1 x + 3 68. y = 4 (-x + 1) 66. 3x + y = 8 y = -3x + 8 67. 2y = _ 2 4 Add or subtract. _ 1. 7m2 + 3m + 4m2 11m2 + 3m y = -4x + 4 Simplify. (Lesson 7-3) 70. cd 4 · (c -5) 3 69. b 4 · b 7 b 11 d _ 4 c 14 71. (-3z 6)2 9z 12 72. (j 3k -5)3 · (k 2)4 7- 7 Adding and Subtracting Polynomials LESSON 7-7 Problem Solving 7-7 PROBLEM SOLVING Adding and Subtracting Polynomials Write the correct answer. 1. There are two boxes in a storage unit. The volume of the first box is 4x3 + 4x2 cubic units. The volume of the second box is 6x3 − 18x2 cubic units. Write a polynomial for the total volume of the boxes. A1NL11S_c07_0504-0509.indd two 509 10x 3 − 14x 2 cubic units 2. The recreation field at a middle school is shaped like a rectangle with a length of 15x yards and a width of 10x − 3 yards. Write a polynomial for the perimeter of the field. Then calculate the perimeter if x = 2. 50x − 6 94 yards LESSON 7-7 Challenge 7-7 CHALLENGE Polynomial Functions A polynomial function is a function whose rule is a polynomial. For example, P(x) = 2x3 4x2 + 3x 2 is a polynomial function. Like other functions, polynomial functions can be evaluated by substituting a value for the variable and simplifying. 4 1. For P(x) = 2x3 4x2 + 3x 2, find P(2). A variable, or variable expression, can be substituted for the variable in a function as well. For example, if P(x) = 5x + 3, then P(a) = 5a + 3. Suppose P(x) = 3x 2. Then P(c) + P(c + 1) = 3c 2 + 3(c + 1) 2 3. Two cabins on opposite banks of a river are 12x2 − 7x + 5 feet apart. One cabin is 9x + 1 feet from the river. The other cabin is 3x2 + 4 feet from the river. Write the polynomial that represents the width of the river where it passes between the two cabins. Then calculate the width if x = 3. = 3c 2 + 3c + 3 2 = 6c 1 3. For P(x) = 10 4x, find P(d) + P(2d) P(3). 22 12d 9x 2 − 16x; 33 feet 4. For P(x) = 3x4 7x3 + 2x2 x + 8, find P(g2) + P(g). The circle graph represents election results for the president of the math team. Use the graph for questions 4–6. Select the best answer. 3g 8 7g 6 g 4 7g 3 + g 2 g + 16 Give an example of Q(x) and S(x) for each situation. 4. The angle value of Greg’s sector can be modeled by x2 + 6x + 2. The angle value of Dion’s sector can be modeled by 7x + 20. Which polynomial represents both sectors combined? A x2 + x + 18 B x2 + 13x + 22 5. Q(x) is a quartic binomial. S(x) is a quartic trinomial. Q(x) + S(x) is a quartic trinomial. Q(x) = x4 + 1; S(x) = x4 + x3 + 1 C 6x2 + 7x + 18 6. Q(x) is a cubic trinomial. S(x) is a cubic binomial. Q(x) S(x) is a quadratic monomial. D 7x2 + 6x + 22 Q(x) = x3 + x2 + 1; S(x) = x3 + 1 5. The sum of Greg and Lynn’s sectors is 2x2 + 4x − 6. The sum of Max and Dion’s sectors is 10x + 26. Which polynomial represents how much greater Greg and Lynn’s combined sectors are than Max and Dion’s? F 2x2 + 6x + 32 G 2x2 − 6x + 20 H 2x2 − 6x − 32 J 2x2 + 14x + 20 6. The sum of Lynn’s sector and Max’s sector is 2x2 − 9x − 2. Max’s sector can be modeled by 3x + 6. Which polynomial represents the angle value of Lynn’s sector? 2 2 A 2x − 6x + 4 C 2x − 12x + 8 B 2x2 − 6x − 4 D 2x2 − 12x − 8 7. Q(x) is a cubic trinomial. S(x) is a cubic binomial. Q(x) + S(x) is a quadratic binomial. k7 509 2. (r2 + s2) - (5r2 + 4s2) -4r2 - 3s2 3. (10pq + 3p) + (2pq - 5p + 6pq) 18pq - 2p 4. (14d 2 - 8) + (6d 2 - 2d + 1) 20d2 - 2d - 7 5. (2.5ab + 14b) (-1.5ab + 4b) 4ab + 10b 7/18/09 5:00:48 PM 4b 3 + b 6 2. For P(x) = 4x3 + x 6, find P(b). j _ 9 6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x2 + 12x + 9, and the area of the second wall is modeled by 36x2 - 12x + 1. Write a polynomial that represents the total area of the two walls. 40x2 + 10 Also available on transparency Q(x) = x3 + x2 + 1; S(x) = x3 + x2 8. Q(x) is a quintic trinomial. S(x) is a quartic trinomial. Q(x) + S(x) is a quintic polynomial with five terms. Q(x) = x5 + x4 + 1; S(x) = x4 + x 3 + x2 Lesson 7-7 509 7-8 Organizer Model Polynomial Multiplication Use with Lesson 7-8 Pacing: 1 Traditional __ day 2 1 __ Block 4 day You can use algebra tiles to multiply polynomials. Use the length and width of a rectangle to represent the factors. The area of the rectangle represents the product. Objective: Use algebra tiles to Use with Lesson 7-8 model polynomial multiplication. GI Materials: algebra tiles < D@<I KEY REMEMBER X Online Edition X X • The product of two values with the same sign is positive. • The product of two values with different signs is negative. Countdown Week 16 Activity 1 Resources Use algebra tiles to find 2(x + 1). Algebra Lab Activities 7-8 Lab Recording Sheet MODEL Teach ÝÊÊ£ Discuss In Activity 3, remind students that zero pairs are only those whose size and shape are the same, but whose colors are different. Place the first factor in a column along the left side of the grid. This will be the width of the rectangle. 2(x + 1) Place the second factor across the top of the grid. This will be the length of the rectangle. Ó In Activity 1, show students that they would get the same product if they placed the first factor along the top and the second factor along the left side. This is because multiplication is commutative. ALGEBRA Fill in the grid with tiles that have the same width as the tiles in the left column and the same length as the tiles in the top row. The area of the rectangle inside the grid represents the product. Alternative Approach x+x+1+1 2x + 2 Use the transparency mat and transparency algebra tiles (MK). The rectangle has an area of 2x + 2, so 2(x + 1) = 2x + 2. Notice that this is the same product you would get by using the Distributive Property to multiply 2(x + 1). Try This Use algebra tiles to find each product. 2. 2(2x + 1) 4x + 2 1. 3(x + 2) 3x + 6 510 510 Chapter 7 4. 3(2x + 2) 6x + 6 Chapter 7 Exponents and Polynomials A1NL11S_c07_0510-0511.indd 510 KEYWORD: MA7 Resources 3. 3(x + 1) 3x + 3 6/25/09 9:38:46 AM Close Activity 2 Key Concept Use algebra tiles to find 2x(x - 3). MODEL ÝÊÊÎ ÓÝ When multiplication is being modeled with algebra tiles, the dimensions of a rectangle represent the factors, and the area of the rectangle represents the product. The area of the rectangle can sometimes be simplified by removing zero pairs. ALGEBRA Place tiles to form the length and width of a rectangle and fill in the rectangle. The product of two values with the same sign (same color) is positive (yellow). The product of two values with different signs (different colors) is negative (red). 2x(x - 3) The area of the rectangle inside the grid represents the product. x2 + x2 - x - x - x - x - x - x Assessment The rectangle has an area of 2x 2 - 6x, so 2x(x - 3) = 2x 2 - 6x. Journal When using algebra tiles to model polynomial multiplication, have students explain how to determine which tiles should be placed inside the multiplication grid to form the rectangle (product). Explanations should include how to determine sizes, shapes, and colors of tiles. 2x 2 - 6x Try This Use algebra tiles to find each product. 5. 3x(x - 2) 6. x(2x - 1) 3x 2 - 6x 2x 2 - x 7. x(x + 1) x2+x 8. (8x + 5)(-2x) -16x 2 - 10x Activity 3 Use algebra tiles to find (x + 1)(x - 2). MODEL ÝÊÊÓ ÝÊÊ£ ALGEBRA Place tiles for each factor to form the length and width of a rectangle. Fill in the grid and remove any zero pairs. The area inside the grid represents the product. The remaining area is x 2 - x - 2, so (x + 1)(x - 2) = x 2 - x - 2. (x + 1)(x - 2) x2 - x - x + x - 1 - 1 x2 - x - 1 - 1 x2 - x - 2 Try This Use algebra tiles to find each product. x2-x-6 9. (x + 2)(x - 3) x 2 + 2x - 3 10. (x - 1)(x + 3) x 2 - 5x + 6 11. (x - 2)(x - 3) x 2 + 3x + 2 12. (x + 1)(x + 2) 7- 8 Algebra Lab A1NL11S_c07_0510-0511.indd 511 511 6/25/09 9:39:12 AM 7-8 Algebra Lab 511 7-8 Organizer Multiplying Polynomials 7-8 Pacing: Traditional 1 day 1 Block __ day 2 GI Objective: Multiply polynomials. < D@<I Why learn this? You can multiply polynomials to write expressions for areas, such as the area of a dulcimer. (See Example 5.) Objective Multiply polynomials. Online Edition Tutorial Videos, Interactivity Countdown Week 17 To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter. EXAMPLE Warm Up 1 A (5x 2)(4x 3) Simplify. 1. 32 9 3. 102 Multiplying Monomials Multiply. 2. 24 16 100 4. 23 · 24 6. (53)2 5. y5 · y4 27 7. (x2)4 56 8. -4(x - 7) y9 B x8 -4x + 28 Also available on transparency When multiplying powers with the same base, keep the base and add the exponents. C x2 · x3 = x2+3 = x5 (5x 2)(4x 3) (5 · 4)(x 2 · x 3) Group factors with like bases together. 20x 5 Multiply. (-3x 3y 2)(4xy 5) (-3x 3y 2)(4xy 5) (-3 · 4)(x 3 · x)(y 2 · y 5) Group factors with like bases together. -12x 4y 7 Multiply. (_12 a b)(a c )(6b ) (_12 a b)(a c )(6b ) (_12 · 6)(a · a )(b · b )(c ) 3 2 2 3 2 2 3 2 2 2 2 2 Group factors with like bases together. 3a 5b 3c 2 Student A: What is u times r times r? Multiply. Multiply. 1a. (3x 3)(6x 2) Student B: ur2. 18x Student A: No, I’m not! 5 1b. (2r 2t)(5t 3) 2 4 10r t ( ) 1 x 2y (12x 3z 2) y 4z 5 1c. _ ( ) 3 5 5 7 4x y z To multiply a polynomial by a monomial, use the Distributive Property. EXAMPLE 2 Multiplying a Polynomial by a Monomial Multiply. A 5(2x 2 + x + 4) 5 (2x 2 + x + 4) (5)2x 2 + (5)x + (5)4 10x 2 + 5x + 20 512 Distribute 5. Multiply. Chapter 7 Exponents and Polynomials 1 Introduce A1NL11S_c07_0512-0519.indd e x p l o r512 at i o n 7-8 Motivate Multiplying Polynomials You will need a graphing calculator for this Exploration. As you work through the Exploration, try to ﬁnd a rule for multiplying a monomial and a polynomial. 1. You can use your calculator to explore the relationship between x 2x 2 ⫹ 3 and 2x 3 ⫹ 3x. Press 9 and enter 2 3 x 2x ⫹ 3 as Y1. Then enter 2x ⫹ 3x as Y2. / 2. Press ND '2!0( to view a table of values for the two expressions. Use the arrow keys to scroll up and down the table. What do you notice about the values of Y1 and Y2 for each value of x ? Use this information to 2 3 make a conjecture about x 2x ⫹ 3 and 2x ⫹ 3x. Use a calculator to predict whether each equation is true. 3. 2x 5x ⫹ 9 ⫽ 10x 2 ⫹ 18x KEYWORD: MA7 Resources 4. 3x 3 2x ⫹ 9 ⫽ 6x 4 ⫹ 9 Chapter 7 Have students explain how to find the area of a rectangle with length x and width 10. A = w, so A = (x)(10) = 10x Explain to students that in this lesson they will learn to describe area when the dimensions are polynomials. 5. 4x 2 3x ⫺ 5 ⫽ 12x 3 ⫺ 20x 2 6. 10x 3 3x 2 ⫺ 2x ⫽ 30x 5 ⫺ 20x 4 THINK AND DISCUSS 7. Describe how to multiply a monomial and a polynomial based on your answers to Problems 3–6. 512 Ask students to explain how to find the area of a rectangle with length 5 cm and width 7 cm. A = w, so A = (5)(7) = 35 cm2 Explorations and answers are provided in Alternate Openers: Explorations Transparencies. 6/25/09 9:40:02 AM Multiply. "" B 2x y (3x - y) (2x 2y)(3x - y) (2x 2y)3x + (2x 2y)(-y) When a binomial is raised to a power, many students make the following mistake: (x + 4)2 = x2 + 42. Encourage students to write the expression as the product of two binomials first and then multiply. For example, (x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16. Distribute 2x 2y. (2 · 3)(x 2 · x)y + 2 (-1)(x 2)(y · y) 3 Group like bases together. 6x y - 2x y 2 2 Multiply. C 4a (a 2b + 2b 2) 4a (a 2b + 2b 2) (4a) a 2b + (4a)2b 2 Distribute 4a. (4)(a · a 2)(b) + (4 · 2)(a)(b 2) Group like bases together. 4a b + 8ab Multiply. 3 Ê,,", ,/ 2 2 Additional Examples Example 1 Multiply. 2a. 2 (4x 2 + x + 3) 2b. 3ab (5a 2 + b) 15a 3b + 3ab 2 8x 2 + 2x + 6 2c. 5r 2s 2 (r - 3s) 5r 3s 2 - 15r 2s 3 To multiply a binomial by a binomial, you can apply the Distributive Property more than once: (x + 3)(x + 2) = x (x + 2) + 3(x + 2) Distribute. Multiply. A. (6y3)(3y5) 18y8 B. (3mn2)(9m2n) ( 27m3n3 ) 1 C. _ s2t2 (st)(-12st2) 4 -3s4t5 Example 2 = x (x + 2) + 3(x + 2) Multiply. = x (x) + x (2) + 3 (x) + 3 (2) Distribute again. = x 2 + 2x + 3x + 6 Multiply. = x 2 + 5x + 6 Combine like terms. A. 4(3x2 + 4x - 8) 12x2 + 16x - 32 B. 6pq(2p - q) 12p2q - 6pq2 1 C. _ x2y(6xy + 8x2y2) 2 3x3y2 + 4x4y3 Another method for multiplying binomials is called the FOIL method. F 1. Multiply the First terms. O 2. Multiply the Outer terms. INTERVENTION I Questioning Strategies 3. Multiply the Inner terms. EX AM P LE L 4. Multiply the Last terms. F • When multiplying monomials with exponents, why do you add the exponents? O I EX AM P LE L 7- 8 Multiplying Polynomials 513 2 • Will a monomial times a binomial always be a binomial? • Will a monomial times a trinomial always be a trinomial? Explain. 2 Teach A1NL11S_c07_0512-0519.indd 513 Inclusion When using the FOIL method, encourage students to write the product after drawing each arrow rather than drawing all of the arrows at once. State: “Draw the F arrow and write the product, draw the O arrow and write the product,” and so on. 7/18/09 4:59:16 PM Guided Instruction Review the multiplication properties of exponents before beginning this lesson. Encourage students to write each step carefully to avoid making mistakes when distributing. Arrows can be drawn to help keep track of the terms that have been multiplied. Remind students that all of the methods presented in this lesson will give correct answers. Students should choose the method they are most comfortable with. 1 • How is multiplying monomials different from adding monomials? Through Visual Cues Show students the “FOIL face” to help them keep track of which terms to multiply. ÝÊÊÎ®ÝÊÊÓ® Lesson 7-8 513 EXAMPLE 3 Multiplying Binomials Multiply. Additional Examples A (x + 2)(x - 5) (x + 2 ) (x - 5 ) Example 3 Multiply. A. (s + 4)(s - 2) B. (x - 4)2 s2 + 2s - 8 Distribute. x (x) + x (-5) + 2(x) + 2(-5) Distribute again. x - 5x + 2x - 10 Multiply. x 2 - 3x - 10 Combine like terms. 2 x2 - 8x + 16 C. (8m2 - n)(m2 - 3n) In the expression (x + 5)2, the base is (x + 5). 8m4 - 25m2n + 3n2 B (x + 5) 2 (x + 5)2 = (x + 5)(x + 5) (x + 5)(x + 5) (x · x) + (x · 5) + (5 · x) + (5 · 5) Write as a product of two binomials. x + 5x + 5x + 25 Multiply. x 2 + 10x + 25 Combine like terms. Use the FOIL method. 2 INTERVENTION C (3a 2 - b)(a 2 - 2b) Questioning Strategies EX A M P L E x (x - 5) + 2(x - 5) 3a 2(a 2) + 3a 2(-2b) - b (a 2) - b (-2b) Use the FOIL method. 3 3a 4 - 6a 2b - a 2b + 2b 2 Multiply. 3a - 7a b + 2b Combine like terms. 4 • What terms from FOIL can often be combined? 2 2 3a. (a + 3)(a - 4) • What do the signs from the last terms in each binomial tell you about the signs in the answer? a - a - 12 2 3b. (x - 3)2 x - 6x + 9 2 3c. (2a - b 2)(a + 4b 2) 2a 2 + 7ab 2 - 4b 4 To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x 2 + 10x - 6): Math Background In Example 3C, remind students that the terms 6a2b and -ba2 are like terms due to the Commutative Property of Multiplication. (5x + 3)(2x 2 + 10x - 6) = 5x (2x 2 + 10x - 6) + 3 (2x 2 + 10x - 6) = 5x (2x 2 + 10x - 6) + 3(2x 2 + 10x - 6) = 5x (2x 2) + 5x (10x) + 5x (-6) + 3(2x 2) + 3(10x) + 3(-6) = 10x 3 + 50x 2 - 30x + 6x 2 + 30x - 18 = 10x 3 + 56x 2 - 18 You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x 2 + 10x - 6) and width (5x + 3): 5x 2x 2 + 10x -6 10x 3 50x 2 -30x 30x -18 +3 6x 2 Write the product of the monomials in each row and column. To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary. 10x 3 + 6x 2 + 50x 2 + 30x - 30x - 18 10x 3 + 56x 2 - 18 514 Chapter 7 Exponents and Polynomials Through Modeling A rectangle model can also be used to display the FOIL method. A1NL11S_c07_0512-0519.indd 514 x +3 x x2 3x +2 2x 6 This model shows that (x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6. This rectangle model is useful because it can also be used in the next chapter for factoring polynomials. 514 Chapter 7 6/25/09 9:40:42 AM Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers. Additional Examples 2x 2 + 10x - 6 Example 4 × 5x + 3 −−−−−−−−−−− −−−− 6x 2 + 30x - 18 3 2 + 10x + 50x - 30x −−−−−−−−−−−−−−−−− 10x 3 + 56x 2 + 0x - 18 10x 3 + 56x 2 - 18 EXAMPLE 4 Multiply. Multiply each term in the top polynomial by 3. Multiply each term in the top polynomial by 5x, and align like terms. Combine like terms by adding vertically. Simplify. A. (x - 5)(x2 + 4x - 6) x3 - x2 - 26 x + 30 B. (2x - 5)(-4x2 - 10x + 3) -8x3 + 56 x - 15 C. (x + 3)3 x3 + 9x2 + 27x + 27 D. (3x + 1)(x3 + 4x2 - 7) 3x4 + 13x3 + 4x2 - 21x - 7 Multiplying Polynomials Multiply. A (x + 2)(x 2 - 5x + 4) A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying, has mn terms. In Example 4A, there are 2 · 3, or 6, terms before simplifying. (x + 2)(x 2 - 5x + 4) x (x 2 - 5x + 4) + 2(x 2 - 5x + 4) Distribute. x (x Distribute again. ) + x (-5x) + x (4) + 2(x ) + 2(-5x) + 2(4) 2 2 x 3 + 2x 2 - 5x 2 - 10x + 4x + 8 Simplify. x 3 - 3x 2 - 6x + 8 Combine like terms. INTERVENTION Questioning Strategies EX AM P LE B (3x - 4)(-2x 3 + 5x - 6) (3x - 4)(-2x 3 + 5x - 6) -2x 3 + 0x 2 + 5x - 6 × 3x - 4 −−−−−−−−−−−−−−−− −−−−− 3 2 8x + 0x - 20x + 24 + -6x 4 + 0x 3 + 15x 2 - 18x −−−−−−−−−−−−−−−−−−−−− -6x 4 + 8x 3 + 15x 2 - 38x + 24 4 • How is multiplying a polynomial by a binomial similar to multiplying a binomial by a binomial? How is it different? Add 0x 2 as a placeholder. Multiply each term in the top polynomial by -4. Multiply each term in the top polynomial by 3x, and align like terms. Combine like terms by adding vertically. Critical Thinking Ask students to find similarities between polynomials and real numbers. Discuss the types of operations you can use with each. C (x - 2)3 ⎡⎣(x - 2)(x - 2)⎤⎦(x - 2) Write as the product of three binomials. ⎡⎣x · x + x (-2) + (-2) x + (-2)(-2)⎤⎦(x - 2) Use the FOIL method on the first two factors. (x 2 - 2x - 2x + 4)(x - 2) (x 2 - 4x + 4)(x - 2) (x - 2)(x 2 - 4x + 4) x (x 2 - 4x + 4) + (-2)(x 2 - 4x + 4) x (x 2) + x (-4x) + x (4) + (-2)(x 2) + (-2)(-4x) + (-2)(4) Multiply. Combine like terms. Use the Commutative Property of Multiplication. Distribute. Distribute again. x 3 - 4x 2 + 4x - 2x 2 + 8x - 8 Simplify. x 3 - 6x 2 + 12x - 8 Combine like terms. 7- 8 Multiplying Polynomials A1NL11S_c07_0512-0519.indd 515 515 7/20/09 5:23:11 PM Lesson 7-8 515 Multiply. D (2x + 3)(x 2 - 6x + 5) Additional Examples Example 5 The width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height. x2 - 6x +5 2x 2x 3 -12x 2 10 x +3 3x 2 -18x 15 Write the product of the monomials in each row and column. 2x 3 + 3x 2 - 12x 2 - 18x + 10x + 15 2x 3 - 9x 2 - 8x + 15 Add all terms inside the rectangle. Combine like terms. Multiply. x 3 - x 2 - 6x + 18 4a. (x + 3)(x 2 - 4x + 6) a. Write a polynomial that represents the area of the base of the prism. h2 + h - 12 4b. (3x + 2)(x 2 - 2x + 5) LÓÊÊ ÊÊ£ 3x 3 - 4x 2 + 11x + 10 b. Find the area of the base when the height is 5 feet. 18 ft2 EXAMPLE 5 Music Application A dulcimer is a musical instrument that is sometimes shaped like a trapezoid. L£ÊÊÓ ÊÊ£ A Write a polynomial that represents the area of the dulcimer shown. 1h b +b A=_ ( 1 2) 2 1 h ⎡(2h - 1) + (h + 1)⎤ =_ ⎣ ⎦ 2 1 h (3h) =_ 2 3 _ = h2 2 The area is represented by __32 h 2. INTERVENTION Questioning Strategies EX A M P L E 5 • What types of polynomials did you multiply? Write the formula for area of a trapezoid. Substitute 2h - 1 for b 1 and h + 1 for b 2. Combine like terms. Simplify. B Find the area of the dulcimer when the height is 22 inches. 3 h2 A=_ 2 _ = 3 (22)2 2 3 _ = (484) = 726 2 The area is 726 square inches. 5a. x 2 - 4x Use the polynomial from part a. Substitute 22 for h. 5. The length of a rectangle is 4 meters shorter than its width. a. Write a polynomial that represents the area of the rectangle. b. Find the area of the rectangle when the width is 6 meters. 5b. 12 m2 THINK AND DISCUSS 1. Compare the vertical method for multiplying polynomials with the vertical method for multiplying whole numbers. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, multiply two polynomials using the given method. 516 (y + 2)(y - 5) y2 - 3y - 10 (x + 5y)(xy + 4x + 7y) x2y + 4x2 + 27xy + 5xy2 + 35y2 516 Chapter 7 ÕÌ«Þ}Ê*Þ>Ã ,iVÌ>}i `i 6iÀÌV> iÌ ` Answers to Think and Discuss A1NL11S_c07_0512-0519.indd 516 Ask students to multiply the following using any of the methods learned in this lesson. (5r2s)(9rs) 45r3s2 3x2(x3 - 10) 3x5 - 30x2 " iÌ ` Chapter 7 Exponents and Polynomials 3 Close Summarize ÃÌÀLÕÌÛi *À«iÀÌÞ and INTERVENTION Diagnose Before the Lesson 7-8 Warm Up, TE p. 512 Monitor During the Lesson Check It Out! Exercises, SE pp. 512–516 Questioning Strategies, TE pp. 513–516 Assess After the Lesson 7-8 Lesson Quiz, TE p. 519 Alternative Assessment, TE p. 519 1. Possible answer: Both numbers and polynomials are set up in 2 rows and require you to multiply each item in the top row by an item in the bottom row. In the end, you add vertically to get the answer. When you are multiplying polynomials, the items are monomial terms. When you are multiplying numbers, the items are digits. 2. See p. A7. 6/25/09 9:41:17 AM 7-8 Exercises 7-8 Exercises KEYWORD: MA11 7-8 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide Multiply. SEE EXAMPLE 1 SEE EXAMPLE (-5mn 3)(4m 2n 2) 1 r 4t 3 3. (6rs 2)(s 3t 2) _ 2 4. (_13 a )(12a) 5. (-3x 4y 2)(-7x 3y) 6. (-2pq 3)(5p 2q 2)(-3q 4) 3 8. 3ab (2a 2 + 3b 3) 9. 2a 3b(3a 2b + ab 2) 10. -3x (x - 4x + 6) 11. 5x y (2xy - y) 12. 5m 2n 3 · mn 2(4m - n) 13. (x + 1)(x - 2) 14. (x + 1)2 15. (x - 2)2 16. SEE EXAMPLE 4 2 (y - 3)(y - 5) 17. 19. (x + 5)(x 2 - 2x + 3) 3 18. (m 2 - 2mn)(3mn + n 2) (4a 3 - 2b)(a - 3b 2) 20. (3x + 4)(x 2 - 5x + 2) 21. (2x - 4)(-3x 3 + 2x - 5) 22. (-4x + 6)(2x 3 - x 2 + 1) 23. (x - 5)(x 2 + x + 1) p. 515 SEE EXAMPLE 5 2 p. 514 5 p. 516 Assign Guided Practice exercises as necessary. 2. 7. 4(x 2 + 2x + 1) 2 p. 512 ) (2x 2)(7x 4) p. 512 SEE EXAMPLE ( 1. 24. (a + b)(a - b)(b - a) 25. Photography The length of a rectangular photograph is 3 inches less than twice the width. 2x 2 - 3x a. Write a polynomial that represents the area of the photograph. Homework Quick Check Quickly check key concepts. Exercises: 30, 42, 46, 52, 56, 62, 69 Ý in2 PRACTICE AND PROBLEM SOLVING Multiply. Independent Practice For See Exercises Example 26–34 35–43 44–52 53–61 62 26. 1 2 3 4 5 29. 32. Extra Practice Skills Practice p. S17 Application Practice p. S34 (3x 2)(8x 5) (-2a 3)(-5a) (7x 2)(xy 5)(2x 3y 2) 27. (-2r 3s 4)(6r 2s) 30. (6x 3y 2)(-2x 2y) ( 2 3 2 ) 1 x 2z 3 y 3z 4 28. (15xy 2) _ ( ) 3 31. (-3a 2b)(-2b 3)(-a 3b 2) )(a b c)(3ab 2 9(2x - 5x) 5s 2t 3(2s - 3t 2) -2a 2b 3(3ab 2 - a 2b) 33. (-4a bc 3 ) 4 5 c 34. (12mn )(2m n)(mn) 2 3x(9x - 4x) 2 2 35. 9s(s + 6) 36. 38. 3(2x 2 + 5x + 4) 39. 41. -5x (2x 2 - 3x - 1) 42. 44. (x + 5)(x - 3) 45. (x + 4)2 46. (m - 5)2 47. (5x - 2)(x + 3) 48. (3x - 4)2 49. (5x + 2)(2x - 1) 50. (x - 1)(x - 2) 51. (x - 8)(7x + 4) 52. (2x + 7)(3x + 7) 53. 56. 59. (x + 2)(x 2 - 3x + 5) (x - 3)(x 2 - 5x + 6) (x - 2)(x 2 + 2x + 1) 54. 57. 60. 37. 40. x 2y 3 · 5x 2y (6x + y 2) Answers 1. 14x6 2. -20m3n5 3. 3r5s5t5 4. 4a6 43. -7x 3y · x 2y 2(2x - y) (2x + 5)(x 2 - 4x + 3) (2x 2 - 3)(4x 3 - x 2 + 7) (2x + 10)(4 - x + 6x 3) If you finished Examples 1–5 Basic 26–69, 75–84, 87–89, 98–104 Average 26–65, 70–80 even, 82–93, 98–104 Advanced 26–64, 70–80 even, 82–104 ÓÝÊÊÎ b. Find the area of the photograph when the width is 4 inches. 20 If you finished Examples 1–3 Basic 26–52, 70–78 Average 26–52, 70–78 Advanced 26–52, 70–78, 90 5. 21x7y3 6. 30p3q9 7. 4x2 + 8x + 4 55. (5x - 1)(-2x 3 + 4x - 3) 8. 6a3b + 9ab4 9. 6a5b2 + 2a4b3 58. (x - 4)3 10. -3x3 + 12x2 - 18x 61. (1 - x) 11. 10x3y4 - 5x2y2 3 12. 20m4n5 - 5m3n6 62. Geometry The length of the rectangle at right is 3 feet longer than its width. x 2 + 3x a. Write a polynomial that represents the area of the rectangle. b. Find the area of the rectangle when the width is 5 feet. 40 ft2 13. x2 - x - 2 ÝÊÊÎ 14. x2 + 2x + 1 Ý 63. A square tabletop has side lengths of (4x - 6) units. Write a polynomial that represents the area of the tabletop. 16x 2 - 48x + 36 7- 8 Multiplying Polynomials 15. x2 - 4x + 4 16. y2 - 8y + 15 A1NL11S_c07_0512-0519.indd 17.5174a4 - 2ab - 12a3b2 + 6b3 18. 3m3n - 5m2n2 - 2mn3 19. x3 + 3x2 - 7x + 15 20. 3x3 - 11x2 - 14x + 8 21. + 18x + 20 -6x4 12x3 + 4x2 - 22. -8x4 + 16x3 - 6x2 4x + 6 23. x3 - 4x2 - 4x - 5 24. -a3 + a2b + ab2 - b3 26. 24x7 38. 6x2 + 15x + 12 27. -12r5s5 39. 10s3t3 - 15s2t5 28. 5x3y5z7 29. 10a4 40. 30x5y4 + 5x4y6 41. -10x3 + 15x2 42. -6a3b5 + 2a4b4 31. -6a5b6 43. -14x6y3 + 7x5y4 44. x2 + 2x - 15 33. -12a7b7c8 45. x2 + 8x + 16 32. 34. 24m4n4 35. 9s2 + 54s 36. 18x2 - 45x 37. 27x3 - 12x2 6/25/09 9:41:45 AM + 5x 30. -12x5y3 14x6y7 517 46. m2 - 10m + 25 47. 5x2 + 13x - 6 48. 9x2 - 24x + 16 49–61. See p. A27. KEYWORD: MA7 Resources Lesson 7-8 517 Exercise 64 involves multiplying polynomials. This exercise prepares students for the Multi-Step Test Prep on page 528. 64. This problem will prepare you for the Multi-Step Test Prep on page 528. a. Marie is creating a garden. She designs a rectangular garden with a length of (x + 4) feet and a width of (x + 1) feet. Draw a diagram of Marie’s garden with the length and width labeled. b. Write a polynomial that represents the area of Marie’s garden. x 2 + 5x + 4 c. What is the area when x = 4? 40 ft2 Answers 64a. 65. Copy and complete the table below. x A Degree of A B Degree of B A·B 2x 2 2 3x 5 5 6x 7 3 2x 2 + 1 2 2 x2 - x 2 4 1 x 3 - 2x 2 + 1 3 4 x 65b. x4 - x3 + 2x2 - 2x c. x4 - 5x3 + 6x2 + x - 3 70. 6a9 71. 2x2 - 7x - 30 a. 5x 3 b. x2 + 2 c. x-3 72. 3g2 + 14g - 5 73. 8x2 74. x2 - 16xy + 6y2 75. 6x2 - 9x - 6 66. 77. x3 + 3x2 67. 79. 2x3 - 7x2 - 10x + 24 - - ab2 + 12x 2 + 12x + 3 {Ý ÝÊÊx 8x + 12x Sports b3 ÝÊÊx ÓÝÊÊ£ 2 x 2 - 10x + 25 69. Sports The length of a regulation team handball court is twice its width. a. Write a polynomial that represents the area of the court. 2x 2 81. 8p3 - 36p2q + 54pq2 - 27q3 82a. 5 68. ÎÓÝÊÊ£® ÓÝÊÊÎ 78. x3 + 3x2 + 2x 80. 10x + 5x 3 Geometry Write a polynomial that represents the area of each rectangle. 76. x2 - 6x - 40 a2b 7 5 d. Use the results from the table to complete the following: The product of a polynomial of degree m and a polynomial of degree n has a degree of . m + n -9 a3 Degree of A · B x Ý ÓÝ b. The width of a team handball court is 20 meters. Find the area of the court. 800 m2 x 83. Possible answer: Each letter in FOIL represents a pair of terms in a certain position within the factors. The letters must account for every pairing of terms while describing first, outside, inside, and last positions. This is only possible with 2 binomials. Multiply. Team handball is a game with elements of soccer and basketball. It originated in Europe in the 1900s and was first played at the Olympics in 1936 with teams of 11 players. Today, a handball team consists of seven players—six court players and one goalie. 7-8 PRACTICE C __________________ (1.5a 3)(4a 6) 71. (2x + 5)(x - 6) 72. 73. (4x - 2y)(2x - 3y) 74. (x + 3)(x - 3) 75. (1.5x - 3)(4x + 2) 76. (x - 10)(x + 4) 77. x (x + 3) 78. (x + 1)(x 2 + 2x) 79. (x - 4)(2x 2 + x - 6) 80. 2 (a + b)(a - b)2 81. (3g - 1)(g + 5) (2p - 3q)3 82. Multi-Step A rectangular swimming pool is 25 feet long and 10 feet wide. It is surrounded by a fence that is x feet from each side of the pool. a. Draw a diagram of this situation. b. Write expressions for the length and width of the fenced region. (Hint: How much longer is one side of the fenced region than the corresponding side of the pool?) 2x + 25; 2x + 10 c. Write an expression for the area of the fenced region. 4x 2 + 70x + 250 83. Write About It Explain why the FOIL method can be used to multiply only two binomials at a time. 7-8 PRACTICE A _______________________________________ 70. __________________ Practice B 7-8 PRACTICE B Multiplying Polynomials LESSON 7-8 518 Chapter 7 Exponents and Polynomials Multiply. 1. (6m 4 ) (8m 2 ) 2. (5x 3 ) (4xy 2 ) 3. (10s 5 t)(7st 4 ) 48m 6 20x4 y2 70s6 t 5 2 4. 4(x + 5x + 6) 5. 2x(3x 4) 4x2 + 20x + 24 7. (x + 3) (x + 4) 9. (x 2) (x 5) x2 12x + 36 2x2 + 17x + 30 x2 7x + 10 13. (x + 4) (x 2 + 3x + 5) x3 + 7x2 + 17x + 20 12. (a 2 + b 2 ) (a + b) 5m 4 + m 3 n + 15m + 3n 14. (3m + 4) (m 2 3m + 5) 7-8 READING STRATEGIES Follow a Procedure 2 Multiply (5x − 4) (3x + x − 8). A1NL11S_c07_0512-0519.indd 518 a3 + a 2 b + ab 2 + b 3 15. (2x 5)(4x 2 3x + 1) 3m 3 5m2 + 3m + 20 Reading Strategies There are several methods that can be used to multiply polynomials, depending on the number of terms. There is one procedure that can always be used, no matter how many terms there are. It is shown in the example below. 21x3 y + 28xy2 + 14xy 11. (m 3 + 3) (5m + n) 10. (2x + 5) (x + 6) 7-8 6. 7xy(3x + 4y + 2) 6x2 8x 8. (x 6) (x 6) x2 + 7x + 12 Name _______________________________________ Date __________________ Class__________________ LESSON 2 8x3 26x2 + 17x 5 (5x − 4) (3x2 + x − 8) 1 Use the Distributive Property. 2 Collect like terms. 5x(3x2 + x − 8) −4(3x2 + x − 8) 15x3 + 5x2 − 40x − 12x2 − 4x + 32 15x3 + 5x2 − 40x − 12x2 − 4x + 32 15x3 + (5x2 − 12x2) + (−40x − 4x) + 32 of the rectangle. b. Find the area of the rectangle when the width is 4 inches. 2 w + 3w 28 in2 a. Write a polynomial that represents the area of the rectangle. width is 10 centimeters. 15x3 − 7x2 − 44x + 32 3 Simplify by combining like terms. Use the procedure shown above to answer each of the following. 17. The length of a rectangle is 8 centimeters less than 3 times the width. b. Find the area of the rectangle when the 15x3 + (5x2 − 12x2) + (−40x − 4x) + 32 2 3w 8w 220 cm2 18. Write a polynomial to represent the volume of the rectangular prism. 1. Multiplication was used six times in step 1. How many times would it be used if two binomials were being multiplied? 4 2. In step 2, how do you know that 5x2 and −12x2 are like terms? They have the same exponent on the same variable. 3. In step 3, how do you know the expression is completely simplified? There are no like terms. 1 x3 5 x2 13x + 60 2 2 −6x 5 + 12x 4 − 3x 3 6. (7x + 2) (x − 3) 7x 2 − 19x − 6 518 Chapter 7 Multiplying Polynomials 7-8 RETEACH Multiply (3a 2 b) (4ab3 ). (3a 2 b) (4ab 3 ) (3 • 4) (a 2 • a) (b • b 3 ) Rearrange so that the constants and the variables with the same bases are together. 12a 3 b 4 Multiply. To multiply a polynomial by a monomial, distribute the monomial to each term in the polynomial. 2x(x 2 + 3x + 7) (2x)x 2 + (2x)3x + (2x)7 Distribute. 2x 3 + 6x 2 + 14x Multiply. Multiply. 1. (5x 2 y 3 ) (2xy) 2. (2xyz) (4x 2 yz) 3. (3x) (x 2 y 3 ) 10x3 y4 8x3 y2 z 2 3x3 y3 Fill in the blanks below. Then finish multiplying. 4. 4(x 5) 5. 3x(x + 8) (4 )x (4 )5 (3x )x + (3x )8 4x 20 3x2 + 24x 6. 2x(x 2 6x + 3) (2x )x (2x )6x + (2x )3 2 2x3 12x2 + 6x Multiply. Multiply the polynomials. 4. −3x3(2x2 − 4x + 1) Review for Mastery 7-8 To multiply monomials, multiply the constants, then multiply variables with the same base. Multiply 2x(x 2 + 3x + 7). 16. The length of a rectangle is 3 inches greater than the width. a. Write a polynomial that represents the area LESSON 8. 4x(x 2 + 8) 7. 5(x + 9) 5. (2x + 5) (9x2 + 6x) 18x 3 + 57x 2 + 30x 7. (2x3 + 6x + 8) (x2 − 5x + 1) 2x 5 − 10x 4 + 8x 3 − 22x 2 − 34x + 8 5x + 45 2 10. 3(5 x + 2) 2 3x 21 4x3 32x 3 11. (5a b) (2ab) 4 10a b 2 9. 3x 2 (2x 2 + 5x + 4) 6x4 + 15x3 + 12x2 12. 5y(y 2 + 7y 2) 5y3 + 35y2 10y 6/25/09 9:42:07 AM 84. Geometry Write a polynomial that represents the volume of the rectangular prism. x 3 + 7x 2 + 10x Because the last term of the polynomial in Exercise 87 is negative, the signs of the second terms in the binomials must be different. Eliminate choices A and B. ÝÊÊÓ 85. Critical Thinking Is there any value for x that would make the statement (x + 3)3 = x 3 + 3 3 true? Give an example to justify your answer. Yes; x = 0 Ý ÝÊÊx 86. Estimation The length of a rectangle is 1 foot more than its width. Write a polynomial that represents the area of the rectangle. Estimate the width of the rectangle if its area is 25 square feet. x 2 + x ; 4.5 ft Multiplying the first terms in Exercise 88 gives 2a3, so choices F and J cannot be correct. Students who chose G probably did not distribute fully. 87. Which of the following products is equal to a 2 - 5a - 6? (a - 1)(a - 5) (a - 2)(a - 3) (a + 2)(a - 3) (a + 1)(a - 6) 88. Which of the following is equal to 2a (a 2 - 1)? 2a 2 - 2a 2a 3 - 1 2a 3 - 2a 2a 2 - 1 89. What is the degree of the product of 3x 3y 2z and x 2yz? 5 6 7 10 102–104. See p. A27. Journal Have students explain how to multiply (2x + 3) by (x + 4) using the method of their choice. CHALLENGE AND EXTEND Simplify. 90. 6x 2 4x - 8 - 2(3x 2 - 2x + 4) -x 2 - 6x 91. x - 2x (x + 3) 7x 2 + x ) ( 92. x 4x - 2 + 3x(x + 1) 2 Answers ÝÊÊx 93. The diagram shows a sandbox and the frame that surrounds it. a. Write a polynomial that represents the area of the sandbox. x 2 - 1 ÝÊÊ£ ÝÊÊÎ ÝÊÊ£ Have students count off by fours. Have the 1’s write a monomial, the 2’s write a binomial, the 3’s write a trinomial, and the 4’s write a polynomial of degree four. Rearrange the class into new groups of four, so that each new group has a monomial, a binomial, a trinomial, and a 4th-degree polynomial. (Make some groups of five students if needed.) Have students multiply pairs of their polynomials together. Each group must turn in as many pairs as possible, with work shown for each answer. b. Write a polynomial that represents the area of the frame that surrounds the sandbox. 8x + 16 94. Geometry The side length of a square is (8 + 2x) units. The area of this square is the same as the perimeter of another square with a side length of (x 2 + 48) units. Find the value of x. x = 4 95. Write a polynomial that represents the product of three consecutive integers. Let x represent the first integer. x 3 + 3x 2 + 2x 96. Find m and n so that x m(x n + x n-2 )= x 5 + x 3. Possible answer: m = 2; n = 3 97. Find a so that 2x a(5x 2a-3 + 2x 2a+2) = 10x 3 + 4x 8 a = 2 SPIRAL REVIEW 98. A stop sign is 2.5 meters tall and casts a shadow that is 3.5 meters long. At the same time, a flagpole casts a shadow that is 28 meters long. How tall is the flagpole? (Lesson 2-8) 20 m 7-8 Find the distance, to the nearest hundredth, between each pair of points. (Lesson 5-5) 99. (2, 3) and (4, 6) 3.61 101. (-3, 7) and (-6, -2) 9.49 100. (-1, 4) and (0, 8) 4.12 Multiply. Graph the solutions of each linear inequality. (Lesson 6-5) 102. y ≤ x - 2 103. 4x - 2y < 10 104. -y ≥ -3x + 1 7- 8 Multiplying Polynomials LESSON 7-8 Problem Solving 7-8 PROBLEM SOLVING Multiplying Polynomials Write the correct answer. 1. A bedroom has a length of x + 3 feet and a width of x 1 feet. Write a polynomial to express the area of the bedroom. Then calculate the area if x = 10. 2. The length of a classroom is 4 feet longer than its width. Write a polynomial to express the area of the classroom. Then calculate the area if the width is 22 feet. w2 + 4w A1NL11S_c07_0512-0519.indd x 519 2 + 2x 3 117 square feet 572 square feet 3. Nicholas is determining if he can afford to buy a car. He multiplies the number of months m by i + p + 30f where i represents the monthly cost of insurance, p represents the monthly car payment, and f represents the number of times he fills the gas tank each month. Write the polynomial that Nicholas can use to determine how much it will cost him to own a car both for one month and for one year. i + p + 30f; 12i + 12p + 360f 4. A seat cushion is shaped like a trapezoid. The shorter base of the cushion is 3 inches greater than the height. The longer base is 2 inches shorter than twice the height. Write the polynomial that can be used to find the area of the cushion. (The area of a trapezoid is represented by 1 h(b +b ) . ) 2 1 2 3 h2 + 1 h 2 2 1 The volume of a pyramid can be found by using Bh where B is the 3 area of the base and h is the height of the pyramid. The Great Pyramid of Giza has a square base, and each side is about 300 feet longer than the height of the pyramid. Select the best answer. 5. Which polynomial represents the approximate area of the base of the Great Pyramid? A h + 90,000 B 2h + 90,000 C h2 + 600h + 90,000 D 2h2 + 600h + 90,000 7. The original height of the Great Pyramid was 485 feet. Due to erosion, it is now about 450 feet. Find the approximate volume of the Great Pyramid today. A 562,500 ft3 C B 616,225 ft3 D 99,623,042 ft3 84,375,000 ft3 LESSON 7-8 Challenge 7-8 CHALLENGE The Missing Binomial Determine the missing binomial. Choose from the table below. Each binomial is used once. x+4 x+1 x+1 x−3 x−4 x−6 x+3 x+6 x−2 + 4) (x + 2) = x 2 + 2x + 4x + 8 = x 2 + 6x + 8 2. (x + 1) (x + 5) = x 2 + 5x + x + 5 + = x 2 + 6x + 5 3. (x + 2) (x − 3) = x 2 − 3x + 2x − 6 = x 2 − x − 6 4. (x + 3) (x − 5) = x 2 − 2x − 15 5. (x + 6) (x + 1) = x 2 + 7x + 6 6. (x − 2) (x + 4) = x + 2x − 8 7. (x − 3) (x + 6) = x 2 + 3x − 18 8. (x − 4) (x + 2) = x 2 − 2x − 8 9. (x − 6) (x − 5) = x 2 − 11x + 30 The binomials missing from the following equations are not all listed in the table. Determine the missing binomials. + 2) (x + 7) = x 2 + 9x + 14 11. (x + 3) (x + 4) = x 2 + 7x + 12 12. (x − 6) (x − 2) = x 2 − 8x + 12 13. (x − 4) (x − 10) = x 2 − 14x + 40 14. (x + 6) (x + 3) = x 2 + 9x + 18 15. (x + 1) (x − 4) = x 2 − 3x − 4 F 1 3 h + 200h2 + 30,000h 3 16. (x − 2) (x − 2) = x 2 − 4x + 4 17. (3x − 1) (x + 2) = 3x 2 + 5x − 2 G 1 2 h + 200h + 30,000 3 18. (x − 7) (2x + 3) = 2x 2 − 11x − 21 19. (2x − 3) (3x + 4) = 6x 2 − x − 12 H h3 + 600h2 + 90,000h 2. 4xy2(x + y) 4x2y2 + 4xy3 3. (x + 2)(x - 8) x2 - 6x - 16 4. (2x - 7)(x2 + 3x - 4) 2x3 - x2 - 29x + 28 5. 6mn(m2 + 10mn - 2) 6m3n + 60m2n2 - 12mn 6. (2x - 5y)(3x + y) 6x2 - 13xy - 5y2 10. (x 6. Which polynomial represents the approximate volume of the Great Pyramid? 18s3t3 7/20/09 5:13:33 PM 1. (x 2 519 1. (6s2t2)(3st) 7. A triangle has a base that is 4 cm longer than its height. a. Write a polynomial that represents the area of the 1 triangle. _ h2 + 2h 2 b. Find the area when the height is 8 cm. 48 cm2 Also available on transparency J 3h3 + 600h2 + 90,000h Lesson 7-8 519 Organizer Volume and Surface Area Geometry The volume V of a three-dimensional figure is the amount of space it occupies. The surface area S is the total area of the two-dimensional surfaces that make up the figure. Geometry Pacing: Traditional 1 day 1 Block __ day 2 Objective: Apply polynomial GI operations to finding areas of geometric figures. <D @<I Cylinder ,iVÌ>}Õ>ÀÊ*ÀÃ i r Online Edition h Countdown Week 17 *ÞÀ>` À Ű Ű Ü Ü 6 ŰÜ - ÓŰÜ Ű Ü ® Teach V = πr h S = 2π πr 2 + 2πrh π 2 £ ŰÜ 6Ú Î £ 6ÊÚ ûÀÀ Ó Î Remember Example Students review and apply volume and surface area formulas for geometric figures. Write and simplify a polynomial expression for the volume of the cone. Leave the symbol π in your answer. «£ 1 πr 2h V=_ 3 1 π 6p 2 p + 1 =_ ( )( ) 3 Close Assess The volume of any prism equals the area of the base times the height. Have students explain how to find the volume of the triangular prism in Problem 6. Find the area of the base. It is 2y2 + y. Multiply this by the height, which is 3y. The volume is 6y3 + 3y2. Choose the correct formula. È« Substitute 6p for r and p + 1 for h. 1 π 36p 2 p + 1 =_ ( )( ) 3 Use the Power of a Product Property. 1 (36)π ⎡p 2 p + 1 ⎤ =_ )⎦ ⎣ ( 3 Use the Associative and Commutative Properties of Multiplication. = 12πp 2(p + 1) Distribute 12π p 2. = 12πp 3 + 12πp 2 Try This Write and simplify a polynomial expression for the volume of each figure. 2. 3. 1. L Êx 2k 3 - k 2 - 6k Ó Î Ó Î £Ó ÎL L£ 3b 3 - 12b 2 - 15b ÊÓ Ó 4πn 3 - 16πn 2 + 16πn Write and simplify a polynomial expression for the surface area of each figure. 4. 5. ÓÝ Ý ÊÎ 520 520 Chapter 7 16x + 30x + 6 Ü£ 6. 19y 2 + 14y Þ ÊÎ ÜÎ ÓÞ ÎÞ ÓÞ £ Chapter 7 Exponents and Polynomials A1NL11S_c07_0520.indd 520 KEYWORD: MA7 Resources ÓÝ Ê£ 2 4πw 2 - 4π 6/25/09 9:43:46 AM 7-9 Special Products of Binomials 7-9 Organizer Pacing: Traditional 1 day Block __1 day 2 Objective: Find special products of binomials. Vocabulary perfect-square trinomial difference of two squares GI Why learn this? You can use special products to find areas, such as the area of a deck around a pond. (See Example 4.) Objective Find special products of binomials. <D @<I Online Edition Tutorial Videos Imagine a square with sides of length (a + b): Countdown Week 17 >ÊÊL > > L >Ó >L >L Ó >ÊÊL Warm Up L L Simplify. 1. 42 The area of this square is (a + b)(a + b), or (a + b) . The area of this square can also be found by adding the areas of the smaller squares and rectangles inside. The sum of the areas inside is a 2 + ab + ab + b 2. 2 3. (-2)2 2 2 49 4. (x)2 x2 2 m4 7. 2(6xy) 12xy 8. 2(8x2) 16x2 You can use the FOIL method to verify this: F 4 5. -(5y2) -5y2 6. (m2) This means that (a + b) = a + 2ab + b . 2 2. 72 16 Also available on transparency L (a + b)2 = (a + b)(a + b) = a 2 + ab + ab + b 2 I O = a 2 + 2ab + b 2 A trinomial of the form a 2 + 2ab + b 2 is called a perfect-square trinomial. A perfect-square trinomial is a trinomial that is the result of squaring a binomial. EXAMPLE Finding Products in the Form (a + b) 1 Parent: What happened in math class today? Student: When the teacher said to look for perfect squares, everyone looked at me. 2 Multiply. A (x + 4)2 (a + b)2 = a 2 + 2ab + b 2 Use the rule for (a + b) . (x + 4)2 = x 2 + 2(x)(4) + 4 2 Identify a and b: a = x and b = 4. 2 = x 2 + 8x + 16 B Simplify. (3x + 2y)2 (a + b)2 = a 2 + 2ab + b 2 Use the rule for (a + b) . 2 (3x + 2y)2 = (3x)2 + 2(3x)(2y) + (2y)2 = 9x 2 + 12xy + 4y 2 Identify a and b: a = 3x and b = 2y. Simplify. 7- 9 Special Products of Binomials 521 1 Introduce A1NL11S_c07_0521-0527.indd e 521 x p l o r at i o n 7-9 You can use the FOIL method to ﬁnd the square of a binomial. For example, to ﬁnd a 1 2, ﬁrst write the product as a 1 a 1 . Then use the FOIL method. a 1 2 a 1 a 1 a a11a1 a2 a a 1 a 2 2a 1 2 2 1. Use the FOIL method to complete the table. Power a 1 2 a 2 6/25/09 9:46:19 AM Motivate Special Products of Binomials Expanded Form Product a 1 a 1 a 2 2a 1 2 a 3 2 a 4 2 2. Look for a pattern in the right column of the table. Use the 2 pattern to find a 5 without using the FOIL method. Have students find the products of the following: (x + 3)(x + 3) (x + 4)(x + 4) (x + 5)(x + 5) x2 + 6x + 9 x2 + 8x + 16 x2 + 10x + 25 Discuss with students any patterns they see. Lead them to recognize that the middle term of the trinomial is two times the product of the first and last terms of the binomial. (a + b)(a + b) = a2 + 2ab + b2 3. Find a 9 2 without using the FOIL method. THINK AND DISCUSS 4. Describe a general rule you can use to find a b 2. 5. Show how you can apply your rule to find 7 x 2. Explorations and answers are provided in Alternate Openers: Explorations Transparencies. KEYWORD: MA7 Resources Lesson 7-9 521 Multiply. C Additional Examples (4 + s 2)2 (a + b)2 = a 2 + 2ab + b 2 (4 + s 2)2 = (4)2 + 2(4)(s 2) + (s 2)2 Example 1 Use the rule for (a + b) . 2 Identify a and b: a = 4 and b = s 2. = 16 + 8s 2 + s 4 Multiply. A. (x + 3)2 D (-m + 3)2 x2 + 6x + 9 B. (4s + 3t)2 16s2 + 24st + 9t2 C. (5 + 25 + ) m2 2 10m2 + (a + b)2 = a 2 + 2ab + b 2 2 Identify a and b: a = -m and b = 3. = m 2 - 6m + 9 Multiply. 1a. (x + 6)2 Multiply. Simplify. 1b. (5a + b) x + 12x + 36 25a + 10ab + b 1c. (1 + c 3) 2 2 1 + 2c + c 6 2 You can use the FOIL method to find products in the form (a - b) : 2 x2 - 12x + 36 B. (4m - 10)2 16m2 - 80m + 100 2 F C. (2x - 5y)2 4x2 - 20xy + 25y2 D. (7 - r3)2 Use the rule for (a + b) . (-m + 3)2 = (-m)2 + 2(-m)(3) + 3 2 m4 Example 2 A. (x - 6)2 Simplify. 2 3 L (a - b)2 = (a - b)(a - b) = a 2 - ab - ab + b 2 49 - 14r3 + r6 I O = a 2 - 2ab + b 2 A trinomial of the form a - 2ab + b is also a perfect-square trinomial because it is the result of squaring the binomial (a - b). 2 INTERVENTION EXAMPLE Questioning Strategies EX A M P L E 2 2 Finding Products in the Form (a - b) Multiply. A (x - 5)2 1 (a - b)2 = a 2 - 2ab + b 2 (x - 5)2 = x 2 - 2(x)(5) + 5 2 • How do you find the middle term? • In which position could a perfectsquare trinomial have a term with an odd exponent? EX A M P L E 2 Use the rule for (a - b) . 2 Identify a and b: a = x and b = 5. = x 2 - 10x + 25 Simplify. B (6a - 1)2 (a - b)2 = a 2 - 2ab + b 2 (6a - 1)2 = (6a)2 - 2(6a)(1) + (1)2 2 • How are these examples different from those in Example 1? What causes the differences? How are they the same? = 36 a 2 - 12a + 1 C (4c - 3d) 2 Identify a and b: a = 6a and b = 1. Simplify. 2 (a - b)2 = a 2 - 2ab + b 2 (4c - 3d) 2 = (4c) 2 - 2(4c)(3d) + (3d)2 • Why is the last term of any perfectsquare trinomial always positive? Use the rule for (a - b) . = 16c 2 - 24cd + 9d 2 Use the rule for (a - b) . 2 Identify a and b: a = 4c and b = 3d. Simplify. D (3 - x ) 2 2 (a - b)2 = (a)2 - 2ab + b 2 (3 - x 2)2 = (3) 2 - 2(3)(x 2) + (x 2)2 = 9 - 6x + x 2 2a. x 2 - 14x + 49 2b. 9b 2 - 12bc + 4c 2 2c. a 4 - 8a 2 + 16 522 Multiply. 2a. (x - 7)2 4 Use the rule for (a - b) . 2 Identify a and b: a = 3 and b = x 2. Simplify. 2 2b. (3b - 2c) 2c. (a 2 - 4) 2 Chapter 7 Exponents and Polynomials 2 Teach A1NL11S_c07_0521-0527.indd 522 6/25/09 9:46:42 AM Guided Instruction Review how to multiply any two binomials before starting this section. Show the FOIL 2 2 method for (a + b) and (a - b) next to each other, so students can better visualize the similarities and differences. Tell students that the binomials in this lesson could be multiplied using any of the methods they already know, but that certain types of binomials can be multiplied more quickly knowing these rules. 522 Chapter 7 Through Auditory Cues Have students learn the “verbal rules” for the special products. (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 • first term squared • plus (or minus) two times the product of both terms • plus last term squared Have students create a similar verbal rule for (a + b)(a - b) = a 2 - b 2. You can use an area model to see that (a + b)(a - b) = a 2 - b 2. 1 2 > L >ÊÊL >ÊÊL L > L >ÊÊL Begin with a square with area a 2. Remove a square with area b 2. The area of the new figure is a 2 - b 2. Then remove the smaller rectangle on the bottom. Turn it and slide it up next to the top rectangle. Ê,,", ,/ >ÊÊL L > 3 > "" The new arrangement is a rectangle with length a + b and width a - b. Its area is (a + b)(a - b). When finding the product of (a - b)2, some students might believe that the middle term and the last term will both be negative numbers. Remind them that the last term comes from a number being squared, and thus will never be a negative. So (a + b)(a - b) = a 2 - b 2. A binomial of the form a 2 - b 2 is called a difference of two squares . EXAMPLE 3 Additional Examples Finding Products in the Form (a + b)(a - b) Example 3 Multiply. Multiply. A (x + 6)(x - 6) (a + b)(a - b) = a - b 2 2 (x + 6)(x - 6) = x 2 - 6 2 = x - 36 2 B (x 2 Use the rule for (a + b)(a - b). A. (x + 4)(x - 4) Identify a and b: a = x and b = 6. B. (p2 + 8q)(p2 - 8q) p4 - 64q2 Simplify. + 2y)(x - 2y) C. (10 + b)(10 - b) 2 (a + b)(a - b) = a 2 - b 2 (x + 2y)(x 2 - 2y) = (x 2)2 - (2y)2 = x - 4y Identify a and b: a = x 2 and b = 2y. 2 Simplify. INTERVENTION C (7 + n)(7 - n) (a + b)(a - b) = a 2 - b 2 Use the rule for (a + b)(a - b). (7 + n)(7 - n) = 7 2 - n 2 Identify a and b: a = 7 and b = n. = 49 - n 2 Simplify. Multiply. 3a. (x + 8)(x - 8) 3b. (3 + 2y 2)(3 - 2y 2) x - 64 9 - 4y 2 EXAMPLE 4 100 - b2 Use the rule for (a + b)(a - b). 2 4 x2 - 16 4 Questioning Strategies EX AM P LE 3 • Why is there no middle term in the product (a + b)(a - b)? • Why does the product (a + b)(a - b) always have a minus sign between the terms? 3c. (9 + r)(9 - r) 81 - r 2 Problem-Solving Application A square koi pond is surrounded by a gravel path. Write an expression that represents the area of the path. 1 Understand the Problem The answer will be an expression that represents the area of the path. List the important information: • The pond is a square with a side length of x - 2. • The path has a side length of x + 2. x–2 x+2 7- 9 Special Products of Binomials A1NL11S_c07_0521-0527.indd 523 Math Background The binomials (a + b) and (a - b) are called conjugates. Conjugates are used in many situations because of the special property that their product is always a difference of squares. 523 6/25/09 9:47:01 AM Lesson 7-9 523 2 Make a Plan The area of the pond is (x - 2)2. The total area of the path plus the pond is (x + 2)2. You can subtract the area of the pond from the total area to find the area of the path. Additional Examples Example 4 Write a polynomial that represents the area of the yard around the pool shown below. 10x + 29 3 Solve Step 1 Find the total area. Use the rule for (a + b) : a = x and b = 2. (x + 2)2 = x 2 + 2(x)(2) + 2 2 ÝÊÊx 2 = x 2 + 4x + 4 ÝÊÊÓ Step 2 Find the area of the pond. ÝÊÊÓ Use the rule for (a - b) : a = x and b = 2. 2 (x - 2)2 = x 2 - 2(x)(2) + 2 2 ÝÊÊx = x 2 - 4x + 4 To subtract a polynomial, add the opposite of each term. Step 3 Find the area of the path. area of path = a INTERVENTION Questioning Strategies EX A M P L E = - area of pond x 2 + 4x + 4 - ( x 2 - 4x + 4) total area = x 2 + 4x + 4 - x 2 + 4x - 4 Identify like terms. = (x 2 - x 2) + (4x + 4x) + (4 - 4) Group like terms together. = 8x 4 • What are you finding when you are multiplying the binomials? The area of the path is 8x. • What area are you asked to find? Combine like terms. 4 Look Back • What operation would you need to perform to get this result? Suppose that x = 10. Then one side of the path is 12, and the total area is 12 2, or 144. Also, if x = 10, one side of the pond is 8, and the area of the pond is 8 2, or 64. This means the area of the path is 144 - 64 = 80. According to the solution above, the area of the path is 8x. If x = 10, then 8x = 8(10) = 80. ✓ 4. Write an expression that represents the area of the swimming pool at right. 25 xÊÊÝ xÊÊÝ Ý Ý Special Products of Binomials Perfect-Square Trinomials (a + b) = (a + b)(a + b) = a 2 + 2ab + b 2 2 (a - b)2 = (a - b)(a - b) = a 2 - 2ab + b 2 Difference of Two Squares (a + b)(a - b) = a 2 - b 2 524 Chapter 7 Exponents and Polynomials 3 Close A1NL11S_c07_0521-0527.indd 524 Summarize Have students state whether each product is a perfect-square trinomial or a difference of two squares, and then find the product. (2x + 3y)2 perfect-square trinomial; 4x2 + 12xy + 9y2 (5m + 7) (5m - 7) difference of two squares; 25m2 - 49 (4s - t)2 perfect-square trinomial; 16s2 - 8st + t2 524 Tell students that recognizing special products will greatly help them with factoring in the next chapter. Chapter 7 and INTERVENTION Diagnose Before the Lesson 7-9 Warm Up, TE p. 521 Monitor During the Lesson Check It Out! Exercises, SE pp. 522–524 Questioning Strategies, TE pp. 522–524 Assess After the Lesson 7-9 Lesson Quiz, TE p. 527 Alternative Assessment, TE p. 527 6/25/09 9:47:21 AM Answers to Think and Discuss 1. (a + b)(a - b) = THINK AND DISCUSS a2 - ab + ab - b2 = a2 - b2 1. Use the FOIL method to verify that (a + b)(a - b) = a 2 - b 2. 2. product 2. When a binomial is squared, the middle term of the resulting trinomial is twice the ? of the first and last terms. 3. GET ORGANIZED Copy and complete the graphic organizer. Complete the special product rules and give an example of each. 7-9 3. See p. A7. -«iV>Ê*À`ÕVÌÃÊvÊ>Ã *iÀviVÌ-µÕ>Ài /À>Ã >ÊÊL®ÓÊÊ¶ vviÀiViÊv /ÜÊ-µÕ>ÀiÃ >ÊÊL®ÓÊÊ¶ >ÊÊL®>ÊÊL®ÊÊ¶ Exercises 7-9 Exercises KEYWORD: MA11 7-9 KEYWORD: MA7 Parent GUIDED PRACTICE Assignment Guide 1. Vocabulary In your own words, describe a perfect-square trinomial. Possible answer: a trinomial that is the result of squaring a binomial Multiply. SEE EXAMPLE 1 p. 521 SEE EXAMPLE 2 p. 522 SEE EXAMPLE 2. (x + 7)2 3. (2 + x)2 4. (x + 1)2 5. (2x + 6)2 6. (5x + 9)2 7. 8. (x - 6)2 9. (x - 2)2 11. (8 - x)2 3 p. 523 SEE EXAMPLE 4 p. 523 12. 17. (2x 2 2 13. (7a - 2b) 15. (x + 6)(x - 6) 14. (x + 5)(x - 5) + 3)(2x - 3) 2 18. 16. (5x + 1)(5x - 1) (9 - x )(9 + x ) 3 If you finished Examples 1–2 Basic 21–32, 53–57 Average 21–32, 53–57 Advanced 21–32, 41, 44–48 even, 54, 56 (2a + 7b)2 10. (2x - 1) 2 (6p - q)2 3 19. If you finished Examples 1–4 Basic 21–63, 64, 67–70, 75–82 Average 21–40, 42–52 even, 53–73, 75–82 Advanced 21–52, 58–62, 64–82 (2x - 5y)(2x + 5y) ÝÊÊÎ 20. Geometry Write a polynomial that represents the area of the figure. 2x 2 + 8x + 10 ÝÊÊ£ ÝÊÊ£ ÝÊÊÎ Assign Guided Practice exercises as necessary. Homework Quick Check Quickly check key concepts. Exercises: 22, 28, 36, 39, 42 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 21–26 27–32 33–38 39 1 2 3 4 Extra Practice Skills Practice p. S17 Application Practice p. S34 Multiply. 21. (x + 3)2 24. 27. (p + 2q 3)2 (s 2 - 7)2 22. (4 + z)2 23. 25. (2 + 3x)2 26. 28. 33. (a - 10)(a + 10) 34. - 2)(x + 2) 37. 36. (x 2 (r 2 + 5t)2 29. (a - 8)2 31. (3x - 4)2 30. (5 - w)2 2 (2c - d 3)2 (x 2 + y 2)2 32. (y + 4)(y - 4) (5a 2 + 9)(5a 2 - 9) (1 - x 2)2 35. (7x + 3)(7x - 3) 38. (x 3 + y 2)(x 3 - y 2) 7- 9 Special Products of Binomials Answers 2. x2 + 14x + 49 3. 4 + 4x + x2 A1NL11S_c07_0521-0527.indd 525 + 2x + 1 4. x2 5. 4x2 + 24x + 36 6. 25x2 + 90x + 81 14. x2 - 25 28. 4c2 - 4cd3 + d6 15. x2 - 36 29. a2 - 16a + 64 16. 25x2 17. 4x4 -1 30. 25 - 10w + w2 -9 31. 32. 1 - 2x2 + x4 19. 4x2 - 25y2 33. a2 - 100 x2 + 6x + 9 21. 8. x2 - 12x + 36 22. 16 + 8z + z2 23. x4 + 2x2y2 + 7/18/09 5:26:57 PM - 24x + 16 18. 81 - x6 7. 4a2 + 28ab + 49b2 9. x2 - 4x + 4 9x2 525 34. y2 - 16 35. 49x2 - 9 y4 36. x4 - 4 10. 4x2 - 4x + 1 24. p2 + 4pq3 + 4q6 37. 25a4 - 81 11. 64 - 16x + x2 25. 4 + 12x + 38. x6 - y4 12. 36p2 - 12pq + q2 26. r 4 + 10r2t + 25t2 13. 49a2 - 28ab + 4b2 27. s4 - 14s2 + 49 9x2 KEYWORD: MA7 Resources Lesson 7-9 525 39. πx 2 + 8πx + 16π Answers 64a. 39. Entertainment Write a polynomial that represents the area of the circular puzzle. Remember that the formula for area of a circle is A = πr 2, where r is the radius of the circle. Leave the symbol π in your answer. x 40a. x > 2; 40. values less than or equal to 2 cause the width of the rectangle to be zero or neg., which does not make sense. x 76. x Multi-Step A square has sides that are (x - 1) units long and a rectangle has a length of x units and a width of (x - 2) units. r=x+4 a. What are the possible values of x ? Explain. b. Which has the greater area, the square or the rectangle? square c. What is the difference in the areas? 1 sq. unit Multiply. 77. (x + y)2 x 2 + 2xy + y 2 42. (x - y)2x 2 - 2xy + y 2 43. (x 2 + 4)(x 2 - 4) x 4 - 16 2 2 44. (x 2 + 4) x 4 + 8x 2 + 16 45. (x 2 - 4) x 4 - 8x 2 + 16 46. (1 - x) 2 1 - 2x + x 2 49. x 6 - 2a 3x 3 + a 6 47. (1 + x)2 1 + 2x + x 2 48. (1 - x)(1 + x) 1 - x 2 49. (x 3 - a 3)(x 3 - a 3) y 41. 50. (5 + n)(5 + n) 25 + 10n + n (r - 4t 4)(r - 4t 4) 52. r 2 - 8rt 4 + 16t 8 2 b (a - b)2 a 2 - 2ab + b 2 1 4 (1 - 4) = 9 1 - 2(1)(4) + 4 2 = 9 53. 2 4 4 4 54. 3 2 1 1 a b (a + b)2 a 2 + 2ab + b 2 55. 1 4 25 25 56. 2 5 49 49 57. 3 0 9 9 a b (a + b)(a - b) a2 - b2 58. 1 4 -15 -15 59. 2 3 -5 -5 60. 3 2 5 5 a 78. 36a - 25b 2 Copy and complete the tables to verify the special products of binomials. x 51. (6a - 5b)(6a + 5b) 2 y 2 2 x Math History Beginning about 3000 B.C.E., the Babylonians lived in what is now Iraq and Turkey. Around 575 B.C.E., they built the Ishtar Gate to serve as one of eight main entrances into the city of Babylon. The image above is a relief sculpture from a restoration of the Ishtar Gate. 7-9 PRACTICE A 61. Math History The Babylonians used tables of squares and the formula (a + b)2 - (a - b)2 ab = _____________ to multiply two numbers. Use this formula to find the product 4 35 · 24. 840 62. Critical Thinking Find a value of c that makes 16 x 2 - 24x + c a perfect-square trinomial. c = 9 63. 7-9 PRACTICE C Practice B LESSON 7-9 7-9 PRACTICE B Special Products of Binomials 526 Multiply. 1. (x + 2)2 2. (m + 4)2 x 2 + 4x + 4 2 4. (2x + 5) 5. (3a + 2) 4x 2 + 20x + 25 9a 2 + 12a + 4 2 2 7. (b − 3) 8. (8 − y) 2 11. (4m − 9) 9x 2 − 42x + 49 2 6. (6 + 5b) 36 + 60b + 25b 2 9. (a − 10) 64 − 16y + y 2 2 10. (3x − 7) Name ________________________________________ Date __________________ Class__________________ 9 + 6a + a 2 2 b 2 − 6b + 9 16m 2 − 72m + 81 a 2 − 20a + 100 2 12. (6 − 3n) 36 − 36n + 9n 2 13. (x + 3) (x − 3) 14. (8 + y) (8 − y) 15. (x + 6) (x − 6) x2 − 9 64 − y 2 x 2 − 36 16. (5x + 2) (5x − 2) 17. (10x + 7y) (10x − 7y) 25x 2 − 4 100x 2 − 49y 2 19. Write a simplified expression that represents the... Explain the error below. What is the correct product? (a - b) = a 2 - b 2 Possible answer: The square of a diff. is not the same as a diff. of squares; a 2 - 2ab + b 2. Chapter 7 Exponents and Polynomials 3. (3 + a)2 m 2 + 8m + 16 2 ANALYSIS///// /////ERROR 2 18. (x2 + 3y) (x2 − 3y) x 4 − 9y 2 LESSON 7-9 Reading Strategies 7-9 READING STRATEGIES Use a Concept Map When multiplying a binomial by another binomial, there are two special kinds of products that may result. They are outlined in these concept maps. Definition A1NL11S_c07_0521-0527.indd 526 LESSON 7-9 Review for Mastery 7-9 RETEACH Special Products of Binomials A perfect-square trinomial is a trinomial that is the result of squaring a binomial. (a + b)2 = a2 + 2ab + b2 Add product of 2, a, and b. (a − b)2 = a2 − 2ab + b2 Subtract product of 2, a, and b. Examples (a + b)2 = a2 + 2ab + b2 (n + 52) = n2 + 10n + 25 (a − b)2 = a2 − 2ab + b2 (3x − 2y)2 = 9xy2 − 12xy + 4y2 Multiply (x + 4)2. (x + 4)2 2 x + 2(x)(4) + 4 4 − x2 2 x2 + 8x + 16 Formula a: 4x b: 3 Middle term is added. 32 16x2 − 2(4x)(3) + 32 Middle term is subtracted. 16x2 − 24x + 9 Simplify. Simplify. (a + b) (a − b) = a2 − b2 c. area of the shaded area. State whether each product will result in a perfect-square trinomial. 20. The small rectangle is made larger by adding 2 units to the length and 2 units to the width. a. What is the new area of the smaller rectangle? 16 − x 2 Examples Difference of Squares Non Examples (x + 4) (x + 4) (x2 + y) (x2 − y) = x4 − y2 (y − 3) (y − 3) 1. Which type of special product is x2 − 64y2? difference of squares 2. Which type of special product will result from multiplying (k2 − 3) (k2 − 3)? perfect square trinomial 4. (x + 7)2 Multiply. Then identify the type of special product. 4. (c2 + 10d) (c2 + 10d) c 4 + 20c 2 d + 100d 2 perfect square trinomial 5. (2s + 3) (2s − 3) 4s 2 − 9 difference of squares 6. (2x + 10)2 2 Square a: x Square a: x 14x Square b: 49 2 x + 14x + 49 2x Square b: 1 2 x − 2x + 1 2(a)(b): 2 Square a: x 40x Square b: 100 2 4x + 40x + 100 2(a)(b): Multiply. x 2 − 16x + 64 It will have 3 terms. yes 5. (x − 1)2 7. (x − 8)2 3. Why is (x + 4) (x + 4) not a difference of squares? 3. (5x − 6) (5x − 6) Fill in the blanks. Then write the perfect-square trinomial. 2(a)(b): Answer the following. 2. (x + 2) (x − 2) no 2 20 Chapter 7 1. (x + 5) (x + 5) yes (t + 6) (t − 6) = t2 − 36 b. What is the area of the new shaded area? 526 (4x − 3)2 b: 4 2 b. area of the small rectangle. Multiply (4x − 3)2. a: x a. area of the large rectangle. 36 − x 7/20/09 2:56:08 PM Square b. Square a. Perfect Square Trinomial Formulas Square b. Square a. A trinomial (3 terms) that is the result of squaring a binomial (2 terms) 8. (x + 2)2 x 2 + 4x + 4 9. (7x − 5)2 49x 2 − 70x + 25 Exercise 64 involves finding area using polynomials. This exercise prepares students for the Multi-Step Test Prep on page 528. 64. This problem will prepare you for the Multi-Step Test Prep on page 528. a. Michael is fencing part of his yard. He started with a square of length x on each side. He then added 3 feet to the length and subtracted 3 feet from the width. Make a sketch to show the fenced area with the length and width labeled. b. Write a polynomial that represents the area of the fenced region. x 2 - 9 c. Michael bought a total of 48 feet of fencing. What is the area of his fenced region? 135 ft2 65. Critical Thinking The polynomial ax 2 - 49 is a difference of two squares. Find all possible values of a between 1 and 100 inclusive. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 66. Write About It When is the product of two binomials also a binomial? Explain and give an example. When 1 binomial is in the form a + b and the other is in the form a - b ; (x + 2)(x - 2) = x 2 - 4 67. What is (5x - 6y)(5x - 6y)? 25x 2 - 22xy + 36y 2 25x 2 - 60xy + 36y 2 25x 2 + 22xy + 36y 2 25x 2 + 60xy + 36y 2 68. Which product is represented by the model? (2x + 5)(2x + 5) (5x - 2)(5x - 2) (5x + 2)(5x - 2) (5x + 2)(5x + 2) 69. If a + b = 12 and a 2 - b 2 = 96 what is the value of a? 2 4 8 10 70. If rs = 15 and (r + s)2 = 64, what is the value of r 2 + s 2? 25 30 34 49 ÓxÝÓ £äÝ £äÝ { 3 71. Multiply (x + 4)(x + 4)(x - 4). Have students describe three ways to find the product (x + 2)(x + 2); one method must use the special products rule. 72. Multiply (x + 4)(x - 4)(x - 4). 2 73. If x + bx + c is a perfect-square trinomial, what is the relationship between b and c? 74. You can multiply two numbers by rewriting the numbers as the difference of two squares. For example: 73. b = ±2 √c� 36 · 24 = (30 + 6)(30 - 6) = 30 2 - 6 2 = 900 - 36 = 864 Use this method to multiply 27 · 19. Explain how you rewrote the numbers. 513; rewrite 27 as 23 + 4 and 19 as 23 - 4. SPIRAL REVIEW 75. The square paper that Yuki is using to make an origami frog has an area of 165 cm 2. Find the side length of the paper to the nearest centimeter. (Lesson 1-5) 13 cm 79. 12x 2 + 6x 80. 4m + 6m + 2n + 11 4 For Exercise 69, a2 - b2 = 96 is the difference of two squares, so (a + b)(a - b) = 96, and by substitution, 12(a - b) = 96. After dividing both sides by 12, a - b = 8. This equation and a + b = 12 form a system of linear equations, easily solved by elimination. Journal CHALLENGE AND EXTEND 71. x + 4x 2 16x - 64 72. x 3 - 4x 2 16x + 64 When multiplying binomials of the form (a - b)(a - b) in Exercise 67, the middle term of the product is negative. Choices C and D can be eliminated. 3 81. -3p 3 - 8p Have students create binomials of 2 2 the form (a + b) , (a - b) , and (a + b)(a - b); find each product; and describe how the special products rule is illustrated. Use intercepts to graph the line described by each equation. (Lesson 5-2) 1x+y=4 76. 2x + 3y = 6 77. y = -3x + 9 78. _ 2 7-9 Multiply. Add or subtract. (Lesson 7-7) 79. 3x 2 + 8x - 2x + 9x 2 80. 82. 2t 2 + 16t + 17 81. (2p 3 + p) - (5p 3 + 9p) 82. (8m 4 + 2n - 3m 3 + 6) + (9m 3 + 5 - 4m 4) (12t - 3t 2 + 10) - (-5t 2 - 7 - 4t) 2 x2 + 14x + 49 2 x2 - 4x + 4 1. (x + 7) 2. (x - 2) 2 7- 9 Special Products of Binomials Name _______________________________________ Date __________________ Class__________________ LESSON 7-9 7-9 PROBLEM SOLVING Problem Solving Special Products of Binomials Write the correct answer. 2. A museum set aside part of a large gallery for a special exhibit. 1. This week Kyara worked x + 4 hours. She is paid x 4 dollars per hour. Write a polynomial for the amount that Kyara earned this week. Then calculate her pay if x = 12. Name ________________________________________ Date __________________ Class__________________ 7-9 In Exercises 1–5, use the given values of a and b to complete the table below. $128 3. Gary is building a square table for a kitchen. In his initial sketch, each side measured x inches. After rearranging some furniture, he realized he would have to add one foot to the length and remove one foot from the width and have a rectangular table instead. Write a polynomial to represent the area of the rectangular table. x2 144 in2 F 4x2 + 8 ab 36 81 121 9 4 9 9 169 −60 32 72 112 −160 −15 8 18 28 −40 −4 3 6 4 7 5 −8 0.75x 2 x 65 6. 5. (a 2 + 2ab + b 2 ) − (a 2 − 2ab + b 2 ) = 4ab H 4x2 + 16 ÝÊÊÇ (a + b)2 + (a − b)2 = 2a 2 + 2b 2 8. [(a2 + b2) + (a2 − b2)]2 7. A 3-ft wide path is built around the garden. Which expression represents the area of the path? F 12x + 33 24x + 84 H 4x2 + 28x + 29 J 4x2 + 40x + 100 2 B 0.86x + 3.44x + 28.56 C 7.14x2 + 28.56x + 3.44 D 7.14x2 + 3.44x + 28.56 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-73 ÝÊÊÈ 4ab c. Use algebra to show that your response to part b is true. of (a + b)2 + (a − b)2. G 4x2 - 36xy + 81y2 7. Write a polynomial that represents the shaded area of the figure below. 14x - 85 The difference (a + b)2 − (a − b)2 is four times the product ab. Write a conjecture for the simplification of each sum or difference. Try to do as much of the simplification as you can mentally. 0.86x2 + 28.56x + 3.44 2 6. (m2 + 2n)(m2 - 2n) m4 - 4n2 a. From the table, how are the values of (a + b)2 − (a − b)2 and ab related? b. Based on your work in Exercises 1–5 and your answer to part a, write a simplification of (a + b)2 − (a + b)2. x2 4x + 4 4x2 + 16x + 16 J 4x2 + 8x + 16 4. (2x - 9y) 7/18/09 5:07:17 PM 7. Using a table like the one that you completed in Exercises 1–5 and the values of a and b used there, write a simplification 6. Which polynomial represents the area of the garden outside the fountain? (Use 3.14 for .) A (a + b)2 − (a − b)2 64 −2 5. Which polynomial represents the area of the garden, including the fountain? G (a − b)2 4 4. 4. Which polynomial represents the area of the fountain? D (a + b)2 5 3. 2575 square feet B x2 4x 4 b −3 2. A fountain is in the center of a square garden. The radius of the fountain is x 2 feet. The length of the garden is 2x + 4 feet. Use this information and the diagram for questions 4 7. Select the best answer. C x2 4 a 1. Write a polynomial for the area of the gallery that is not part of the exhibit. Then calculate the area of that section if x = 60. 3. (5x + 2y) 25x2 + 20xy + 4y2 5. (4x + 5y)(4x - 5y) 16x2 - 25y2 The expansions of (a + b)2, (a − b)2, and (a + b) (a − b) show patterns involving a and b. You can discover other patterns involving a and b when you combine these special products. A1NL11S_c07_0521-0527.indd 527 x2 16 A 2x 4 7-9 CHALLENGE Patterns in Special Products Challenge LESSON 527 Holt McDougal Algebra 1 ÝÊÊÇ 9. [(a2 + b2) − (a2 − b2)]2 4a 4 4b 4 Simplify each expression. 10. [(a2 + b2) + (a2 − b2)]2 ÝÊÊÈ 11. [(a2 + b2) − (a2 − b2)]2 [(a 2 + b 2 ) + (a 2 − b 2 )] 2 = (a 2 + a 2 + b 2 − b 2 ) 2 = (2a 2 ) 2 = 22a 4 = 4a 4 [(a 2 + b 2 ) − (a 2 − b 2 )] 2 = (a 2 − a 2 + b 2 + b 2 ) 2 = (2b 2 ) 2 = 22b 4 = 4b 4 Also available on transparency Lesson 7-9 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 7-72 Holt McDougal Algebra 1 527 SECTION 7B SECTION 7B Polynomials Don’t Fence Me In James has 500 feet of fencing to enclose a rectangular region on his farm for some sheep. Organizer 1. Make a sketch of three possible regions that James could enclose and give the corresponding areas. Objective: Assess students’ GI ability to apply concepts and skills in Lessons 7-6 through 7-9 in a real-world format. <D @<I 2. If the length of the region is x, find an expression for the width. 250 - x 3. Use your answer to Problem 2 to write an equation for the area of the region. A = x (250 - x) Online Edition 4. Graph your equation from Problem 3 on your calculator. Sketch the graph. 5. James wants his fenced region to 1. Possible answers: have the largest area possible using Resources Algebra I Assessments A = 15,625 ft 2 www.mathtekstoolkit.org Problem Text Reference 1 Lesson 7-7 2 Lesson 7-6 3 Lesson 7-8 4 Lesson 7-9 5 Lesson 7-9 6 Lesson 7-9 FT 500 feet of fencing. Find this area using the graph or a table of values. 15,625 ft 2 6. What are the length and width of the region with the area from Problem 5? Describe this region. 125 ft × 125 ft; square FT A = 10,000 ft 2 FT FT A = 15,000 ft 2 FT FT 4. 528 Chapter 7 Exponents and Polynomials INTERVENTION Scaffolding Questions A1NL11S_c07_0528-0529.indd 528 1. What is the perimeter of a rectangle? distance around the rectangle How is it calculated? P = 2 + 2w 2. What are some possible values for and w? Possible answers: = 40, w = 210; = 30, w = 220; = 45, w = 205 3. What is the area of a rectangle? number of square units inside the rectangle How is it calculated? A = w KEYWORD: MA7 Resources 528 Chapter 7 4. What is an appropriate viewing window? Possible answer: 0 < x < 250 and 0 < y < 16,000 5. How can you find the greatest area from the graph or table? Graph: find the ycoordinate of the highest point; Table: find the greatest y-value. 6. How can you find the dimensions from the graph or table? Graph: find the x-coordinate of the highest point; Table: look for the greatest y-value and find its corresponding x-value. Extension Now James will use his fencing to create three sides of a rectangular region and build a wall for the fourth side. Now what is the greatest possible area? Describe this region. What is the minimum length of the wall? 31,250 ft2; 125 ft × 250 ft rectangle; 250 ft 6/25/09 9:49:40 AM SECTION 7B SECTION 1. 2r 6 + 4r 2 - 3r ; 2 2. -8y 3 + y 2 + 2y + 7; -8 3. t 4 - 12t 3 - 4t ; 1 Quiz for Lessons 7-6 Through 7-9 7-6 Polynomials 7B Write each polynomial in standard form and give the leading coefficient. 1. 4r 2 + 2r 6 - 3r 2. y 2 + 7 - 8y 3 + 2y 4. n + 3 + 3n 3n + n + 3; 3 5. 2 + 3x 2 2 3 3x + 2; 3 6. -3a 2 + 16 + a 7 + a Classify each polynomial according to its degree and number of terms. 7. 2x 3 + 5x - 4 a 7 - 3a 2 + a + 16; 1 Objective: Assess students’ mastery of concepts and skills in Lessons 7-6 through 7-9. 9. 6p 2 + 3p - p 4 + 2p 3 8. 5b 2 10. x 2 + 12 - x Organizer 3. -12t 3 - 4t + t 4 3 11. -2x 3 - 5 + x - 2x 7 12. 5 - 6b 2 + b - 4b 4 13. Business The function C(x) = x 3 - 15x + 14 gives the cost to manufacture x units of a product. What is the cost to manufacture 900 units? $728,986,514 Resources Assessment Resources 7-7 Adding and Subtracting Polynomials Section 7B Quiz Add or subtract. 14. 16. 18. (10m + 4m ) + (7m + 3m) (12d 6 - 3d 2 ) + (2d 4 + 1) (7n 2 - 3n) - (5n 2 + 5n) 3 2 15. (3t - 2t) + (9t + 4t - 6) 2 2 2 17. (6y 3 + 4y 2 ) - (2y 2 + 3y) Test & Practice Generator 19. (b - 10) - (-5b + 4b) 2 3 20. Geometry The measures of the sides of a triangle are shown as polynomials. Write a simplified polynomial to represent the perimeter of the triangle. 2s 3 + 4s 2 + 5s + 5 ÓÃÎÊÊ{ INTERVENTION xÃ Resources {ÃÓÊÊ£ Ready to Go On? Intervention and Enrichment Worksheets 7-8 Multiplying Polynomials Multiply. 21. 2h 3 · 5h 5 22. (s 8 t 4 )(-6st 3 ) 23. 2ab (5a 3 + 3a 2 b) (3k + 5)2 25. (2x 3 + 3y)(4x 2 + y) 26. 24. Ready to Go On? CD-ROM (p 2 + 3p)(9p 2 - 6p - 5) Ready to Go On? Online 27. Geometry Write a simplified polynomial expression for the area of a parallelogram whose base is (x + 7) units and whose height is (x - 3) units. x 2 + 4x - 21 square units ( ) 7-9 Special Products of Binomials Multiply. (d + 9)2 2 32. (a - b) 28. (2x + 5y)2 29. (3 + 2t)2 30. 33. (3w - 1)2 34. (c + 2)(c - 2) 31. (m - 4)2 35. (5r + 6)(5r - 6) 36. Sports A child’s basketball has a radius of (x - 5) inches. Write a polynomial that represents the surface area of the basketball. (The formula for the surface area of a sphere is S = 4πr 2 , where r represents the radius of the sphere.) Leave the symbol π in your answer. 4πx 2 - 40πx + 100π in 2 ( ) Answers 7–12, 14–19. See p. A27. Ready to Go On? NO A1NL11S_c07_0528-0529.indd 529 READY Ready to Go On? Intervention TO Worksheets 21–26, 28–35. See pp. A27–A28. YES Diagnose and Prescribe INTERVENE 529 ENRICH 6/25/09 9:50:08 AM GO ON? Intervention, Section 7B CD-ROM Lesson 7-6 7-6 Intervention Activity 7-6 Lesson 7-7 7-7 Intervention Activity 7-7 Lesson 7-8 7-8 Intervention Activity 7-8 Lesson 7-9 7-9 Intervention Activity 7-9 Online Diagnose and Prescribe Online READY TO GO ON? Enrichment, Section 7B Worksheets CD-ROM Online Ready to Go On? 529 CHAPTER Study Guide: Review 7 Organizer Vocabulary index . . . . . . . . . . . . . . . . . . . . . . 488 quadratic . . . . . . . . . . . . . . . . . . 497 cubic . . . . . . . . . . . . . . . . . . . . . . 497 leading coefficient . . . . . . . . . . 497 scientific notation . . . . . . . . . . 467 organize and review key concepts and skills presented in Chapter 7. degree of a monomial . . . . . . . 496 monomial . . . . . . . . . . . . . . . . . . 496 degree of a polynomial . . . . . . 496 perfect-square trinomial . . . . 521 standard form of a polynomial . . . . . . . . . . . . . . 497 difference of two squares . . . . 523 polynomial . . . . . . . . . . . . . . . . . 496 trinomial. . . . . . . . . . . . . . . . . . . 497 GI binomial . . . . . . . . . . . . . . . . . . . 497 Objective: Help students <D @<I Online Edition Multilingual Glossary Complete the sentences below with vocabulary words from the list above. 1. A(n) ? polynomial is a polynomial of degree 3. −−−−−− 2. When a polynomial is written with the terms in order from highest to lowest degree, it is in ? . −−−−−− 3. A(n) ? is a number, a variable, or a product of numbers and variables with −−−−−− whole-number exponents. Countdown Week 17 Resources 4. A(n) ? is a polynomial with three terms. −−−−−− 5. ? is a method of writing numbers that are very large or very small. −−−−−− PuzzleView Test & Practice Generator Multilingual Glossary Online 7-1 Integer Exponents (pp. 460– 465) KEYWORD: MA7 Glossary EXAMPLES 6. The diameter of a certain bearing is 2 -5 in. Simplify this expression. Simplify. Lesson Tutorial Videos CD-ROM ■ -2 -4 1 = -_ 1 1 -2 -4 = - _ = -_ 2·2·2·2 16 24 Answers ■ 1. cubic 30 30 = 1 2. standard form of a polynomial ■ 3. monomial 4. trinomial 5. scientific notation 1 6. _ in. 32 7. 1 8. 1 1 9. _ 125 1 10. _, or 0.0001 10,000 1 11. _ 16 1 12. _ 256 27 13. _ 4 1 _ 14. 2 m 15. b 1 16. - _ 2x2y4 17. 2b6c4 3a2 18. _ 4c2 s3 19. _2 qr 530 Chapter 7 ■ Simplify Simplify. 7. (3.6) 0 9. 5 Any nonzero number raised to the zero power is 1. Evaluate r 3s -4 for r = -3 and s = 2. r 3s -4 (-3)(-3)(-3) 27 (-3) 3(2) -4 = __ = - _ 16 2·2·2·2 -3 4 a b _ . -3 4 c -2 4 2 b c a b =_ _ c -2 a3 530 EXERCISES -3 8. (-1) -4 10. 10 -4 Evaluate each expression for the given value(s) of the variable(s). 11. b -4 for b = 2 ( ) 2b 12. _ 5 -4 for b = 10 13. -2p 3q -3 for p = 3 and q = -2 Simplify. 14. m -2 15. bc 0 1 x -2y -4 16. - _ 2 2b 6 17. _ c -4 3a 2c -2 18. _ 4b 0 q -1r -2 19. _ s -3 Chapter 7 Exponents and Polynomials A1NL11S_c07_0530-0535.indd 530 6/25/09 2:50:06 PM Answers 7-2 Powers of 10 and Scientific Notation (pp. 466– 471) ■ Write 1,000,000 as a power of 10. 1,000,000 The decimal point is 6 places 1,000,000 = 10 6 to the right of 1. Find the value of 386.21 × 10 5. 386. 2 1 0 0 0 Move the decimal point 5 places to the right. 38,621,000 ■ 21. 0.00001 EXERCISES EXAMPLES ■ 20. 10,000,000 Write 0.000000041 in scientific notation. 0.0 0 0 0 0 0 0 4 1 Move the decimal point 8 places to the right to get a number between 1 and 10. 4.1 × 10 -8 22. 102 Find the value of each power of 10. 20. 10 7 21. 10 -5 23. 10-11 24. 325,000 Write each number as a power of 10. 22. 100 23. 0.00000000001 25. 1800 Find the value of each expression. 24. 3.25 × 10 5 25. 0.18 × 10 4 27. 0.000299 26. 17 × 10 -2 26. 0.17 28. 5.8 × 10-7, 6.3 × 10-3, 2.2 × 102, 1.2 × 104 27. 299 × 10 -6 29. $38,500,000,000 28. Order the list of numbers from least to greatest. 6.3 × 10 -3, 1.2 × 10 4, 5.8 × 10 -7, 2.2 × 10 2 30. 59 31. 23 · 34 29. In 2003, the average daily value of shares traded on the New York Stock Exchange was about $3.85 × 10 10. Write this amount in standard form. 32. b10 33. r5 34. x12 Simplify. 30. 5 3 · 5 6 31. 2 6 · 3 · 2 -3 · 3 3 The powers have the same base. 32. b · b 33. r 4 · r 35. 1 1 , or 36. _ 23 1 37. _ , or 54 1 38. _6 16b 39. g12h8 Add the exponents. 34. (x 3) 35. (s 3) 40. x4y2 7-3 Multiplication Properties of Exponents (pp. 474– 480) EXERCISES EXAMPLES Simplify. ■ ■ 5 3 · 5 -2 5 3 · 5 -2 5 3 +(-2) 51 5 -3 36. (2 (a 4 · a -2) · (b -3 · b) a 2 · b -2 a2 _ b2 ■ (a -3b 2) -2 (a -3) -2 · (b 2) -2 Use properties to group factors. Add the exponents of powers with the same base. Write with a positive exponent. Power of a Product Property a 6 · b -4 Power of a Power Property a6 _ b4 Write with a positive exponent. 8 4 37. (5 ) 3 -1 38. (4b 3) -2 a ·b ·b·a a 4 · b -3 · b · a -2 4 2 41. -x4y2 ) 2 -2 39. (g 3h 2) -2 40. (-x y) 2 0 2 42. (x 2y 3)(xy 43. j6k9 1 44. _ 5 45. m8n30 2 43. (j 2k 3)(j 4k 6) ) 3 4 44. (5 3 · 5 -2) 42. x6y15 4 41. -(x y) 2 45. (mn 3) (mn 5) -1 5 46. 8 × 1011 3 46. (4 × 10 8)(2 × 10 3) 47. (3 × 10 2)(3 × 10 5) 48. (5 × 10 3 49. (7 × 10 5 50. (3 × 10 -4 51. (3 × 10 -8 )(2 × 10 ) 6 )(2 × 10 ) 5 47. 9 × 107 48. 1 × 1010 )(4 × 10 ) 9 49. 2.8 × 1015 )(6 × 10 ) 50. 6 × 101 -1 51. 1.8 × 10-8 52. In 2003, Wyoming’s population was about 5.0 × 10 5. California’s population was about 7.1 × 10 times as large as Wyoming’s. What was the approximate population of California? Write your answer in scientific notation. Study Guide: Review A1NL11S_c07_0530-0535.indd 531 1 _ 8 1 _ 625 52. 3.55 × 107 531 6/25/09 7:07:09 PM Study Guide: Review 531 Answers 7-4 Division Properties of Exponents (pp. 481– 487) 53. 64 54. m5 7 55. _ 32 56. 6b EXERCISES EXAMPLES x Simplify _. 9 ■ Simplify. x2 x 9 = x 9-2 = x 7 _ x2 57. t3v4 Subtract the exponents. 58. 16 59. 5 × 101 60. 2.5 × 107 28 53. _ 22 m6 54. _ m 26 · 4 · 73 55. _ 25 · 44 · 72 24b 6 56. _ 4b 5 t 4v 5 57. _ tv 1 58. _ 2 () -4 61. 9 Simplify each quotient and write the answer in scientific notation. 62. 7 59. (2.5 × 10 8) ÷ (0.5 × 10 7) 63. 16 60. (2 × 10 10) ÷ (8 × 10 2) 64. 8 65. z 2 7-5 Rational Exponents (pp. 488– 493) 66. 5x 2 4 3 67. x y ■ 69. 0 6 12 Simplify each expression. 1 _ r 6 s 12 3 √r s = ( 6 12 ) 1 _ Definition of 1 _ = (r 6) 3 · (s 12) 3 71. 6 ( 72. 1 73. Simplify √r s . 3 3 70. 4 3n2 EXERCISES EXAMPLES 68. m 2n 4 = r + 2n - 4; 3 ) · (s ) 1 6·_ 3 74. -a6 - a4 + 3a3 + 2a; -1 = (r 2) · (s 4) 75. linear binomial =r s 1 12 · _ 3 Power of a Product Property Power of a Power Property Simplify exponents. 61. 81 2 62. 343 3 2 _ 64. (2 6) 2 Simplify each expression. All variables represent nonnegative numbers. 5 65. √ z 10 67. 76. quintic monomial 1 _ 63. 64 3 2 4 77. quartic trinomial 1 _ 1 _ 1 _ bn 3 66. √ 125 x 6 √ x 8y 6 3 68. √ m 6n 12 7-6 Polynomials (pp. 496– 501) 78. constant monomial EXERCISES EXAMPLES ■ ■ Find the degree of the polynomial 3x + 8x . 3x 2 + 8x 5 8x 5 has the highest degree. Find the degree of each polynomial. 69. 5 70. 8st 3 The degree is 5. 71. 3z 6 Classify the polynomial y 3 - 2y according to its degree and number of terms. Degree: 3 Terms: 2 Write each polynomial in standard form. Then give the leading coefficient. 73. 2n - 4 + 3n 2 74. 2a - a 4 - a 6 + 3a 3 The polynomial y 3 - 2y is a cubic binomial. Classify each polynomial according to its degree and number of terms. 2 5 72. 6h 75. 2s - 6 76. -8p 5 77. -m - m - 1 4 532 Chapter 7 78. 2 Chapter 7 Exponents and Polynomials A1NL11S_c07_0530-0535.indd 532 532 2 7/20/09 5:17:24 PM Answers 7-7 Adding and Subtracting Polynomials (pp. 504– 509) Add. (h 3 - 2h) + (3h 2 + 4h) - 2h 3 81. 3h3 - 3h2 + 5 Add or subtract. 79. 3t + 5 - 7t - 2 82. 2m2 - 5m - 1 83. p2 + 5p + 8 (h 3 - 2h) + (3h 2 + 4h) - 2h 3 (h 3 - 2h 3) + (3h 2) + (4h - 2h) 80. 4x 5 - 6x 6 + 2x 5 - 7x 5 -h 3 + 3h 2 + 2h 82. (3m - 7) + (2m 2 - 8m + 6) (n 3 + 5 - 6n 2) - (3n 2 - 7) (n + 5 - 6n ) + (-3n + 7) (n 3 + 5 - 6n 2) + (-3n 2 + 7) n 3 + (-6n 2 - 3n 2) + (5 + 7) 3 84. -7z2 - z + 10 81. -h 3 - 2h 2 + 4h 3 - h 2 + 5 85. 3g2 + 2g + 4 86. -x2 + 4x + 8 83. (12 + 6p) - (p - p + 4) 2 Subtract. ■ 80. -6x6 - x5 EXERCISES EXAMPLES ■ 79. -4t + 3 2 2 87. 8r2 84. (3z - 9z 2 + 2) + (2z 2 - 4z + 8) 88. 6a6b 85. (10g - g 2 + 3) - (-4g 2 + 8g - 1) 89. 18x3y2 92. 90. 3s6t14 (-5x 3 + 2x 2 - x + 5) - (-5x 3 + 3x 2 - 5x - 3) 91. 2x2 - 8x + 12 n 3 - 9n 2 + 12 92. -3a2b2 + 6a3b2 - 15a2b 93. a2 - 3a - 18 94. b2 - 6b - 27 7-8 Multiplying Polynomials (pp. 512– 519) EXAMPLES Multiply. ■ (2x - 4)(3x + 5) 2x(3x) + 2x(5) - 4(3x) - 4(5) 6x 2 + 10x - 12x - 20 6x 2 - 2x - 20 ■ (b - 2)(b 2 + 4b - 5) b(b 2) + b(4b) - b(5) - 2(b 2) - 2(4b) - 2(-5) b 3 + 4b 2 - 5b - 2b 2 - 8b + 10 b 3 + 2b 2 - 13b + 10 95. x2 - 12x + 20 EXERCISES 96. t2 - 1 97. 8q2 + 34q + 30 Multiply. 87. (2r)(4r) 88. (3a )(2ab) 89. (-3xy)(-6x 2y) 1 s 2t 8 90. (3s 3t 2)(2st 4) _ 2 91. 2(x 2 - 4x + 6) 92. -3ab(ab - 2a 2b + 5a) 101. m2 + 12m + 36 93. (a + 3)(a - 6) 94. (b - 9)(b + 3) 102. 9c2 + 42c + 49 95. (x - 10)(x - 2) 96. (t - 1)(t + 1) 97. (2q + 6)(4q + 5) 98. (5g - 8)(4g - 1) 98. 20g2 - 37g + 8 5 ( ) 99. p2 - 8p + 16 100. x2 + 24x + 144 103. 4r2 - 4r + 1 104. 9a2 - 6ab + b2 105. 4n2 - 20n + 25 106. h2 - 26h + 169 107. x2 - 1 7-9 Special Products of Binomials (pp. 521–527) EXAMPLES Multiply. ■ 109. c4 - d 2 EXERCISES 110. 9k4 - 49 Multiply. (2h - 6)2 (2h - 6) 2 = (2h) 2 + 2(2h)(-6) + (-6) 2 99. (p - 4) 2 100. (x + 12) 2 101. (m + 6)2 102. (3c + 7)2 4h - 24h + 36 103. (2r - 1) 2 2 104. (3a - b) (4x - 3)(4x + 3) (4x - 3)(4x + 3) = (4x) 2 - 3 2 105. (2n - 5)2 2 106. (h - 13) 107. (x - 1)(x + 1) 108. (z + 15)(z - 15) 16x 2 - 9 109. (c - d)(c + d) 110. (3k 2 + 7)(3k 2 - 7) 2 ■ 108. z2 - 225 2 2 Study Guide: Review A1NL11S_c07_0530-0535.indd 533 533 7/18/09 5:04:22 PM Study Guide: Review 533 CHAPTER 7 Organizer Evaluate each expression for the given value(s) of the variable(s). ( ) 1b 1. _ 3 Objective: Assess students’ Simplify. GI mastery of concepts and skills in Chapter 7. @<I <D -2 1 _ for b = 12 2. (14 - a 0b 2) 16 _ _2 3. 2r -3 r for a = -2 and b = 4 - _ 2 5. m 2n -3 m 4. -3f 0g -1 - 3 g 3 -3 n 8 t _ 3 1 s -5t 3 6. _ 2 3 _1 2s 5 Write each number as a power of 10. Online Edition 7. 0.0000001 10-7 8. 10,000,000,000,000 10 13 9. 1 10 0 Find the value of each expression. Resources 10. 1.25 × 10 -5 0.0000125 Assessment Resources 11. 10 8 × 10 -11 0.001 12. 325 × 10 -2 3.25 Chapter 7 Tests 13. Technology In 2002, there were approximately 544,000,000 Internet users worldwide. Write this number in scientific notation. 5.44 × 10 8 • Free Response (Levels A, B, C) Simplify. 14. (f • Multiple Choice (Levels A, B, C) ) 4 3 15. (4b 2) f 12 0 16. (a 3b 6) 1 6 17. -(x 3) · (x 2) 5 a 18b 36 6 -x 27 Simplify each quotient and write the answer in scientific notation. • Performance Assessment 18. (3.6 × 10 9) ÷ (6 × 10 4) 6 × 10 4 19. (3 × 10 12) ÷ (9.6 × 10 16) 3.125 × 10-5 Simplify. Test & Practice Generator _ y4 3 20. _ y y _ d 2f 5 f9 21. _ 2 (d 3) f -4 d 4 1 ( ) · (_2s6t ) _ 16 25 · 33 · 54 3 22. _ 28 · 32 · 54 8 4s 23. _ 3t -2 2 24. Geometry The surface area of a cone is approximated by the polynomial 3.14r 2 + 3.14r, where r is the radius and is the slant height. Find the approximate surface area of a cone when = 5 cm and r = 3 cm. 75.36 cm 2 Simplify each expression. All variables represent nonnegative numbers. ( ) _35 27 25. _ 125 1 _ 3 3 26. √ 43 3 43 4 √ 25y 8 5y -10b 3 - 6b 2 Add or subtract. 29. 3a - 4b + 2a 5a - 4b 27. 30. (2b - 4b 2 ) - (6b 3 3 + 8b 2 ) Multiply. 32. -5(r s - 6) -5r s + 30 33. (2t - 7)(t + 4) 2t + t - 28 35. (m + 6) 2 m 2 + 12m + 36 36. (3t - 7)(3t + 7) 9t 2 - 49 2 2 2 5 28. √ 3 5 t 10 3t 2 -4g 3 - 6g 2 + g - 4 31. -9g 2 + 3g - 4g 3 - 2g + 3g 2 - 4 16g 3 - 24g 2 - 7g + 3 34. (4g - 1)(4g 2 - 5g - 3) 37. (3x 2 - 7) 2 9x 4 - 42x 2 + 49 38. Carpentry Carpenters use a tool called a speed square to help them mark right angles. A speed square is a right triangle. a. Write a polynomial that represents the area of the speed square shown. b. Find the area when x = 4.5 in. 3.75 in 2 534 534 Chapter 7 ÓÝÊÊÈ Chapter 7 Exponents and Polynomials A1NL11S_c07_0530-0535.indd 534 KEYWORD: MA7 Resources x 2 - x - 12 ÝÊÊ{ Ê£ÊÊL ÊÊÚ Ó 7/18/09 5:04:26 PM CHAPTER 7 Organizer FOCUS ON SAT When you receive your SAT scores, you will find a percentile for each score. The percentile tells you what percent of students scored lower than you on the same test. Your percentile at the national and state levels may differ because of the different groups being compared. You may want to time yourself as you take this practice test. It should take you about 7 minutes to complete. 1. If (x + 1)(x + 4) - (x - 1)(x - 2) = 0, what is the value of x? Objective: Provide practice for You may use some types of calculators on the math section of the SAT. For about 40% of the test items, a graphing calculator is recommended. Bring a calculator that you are comfortable using. You won’t have time to figure out how a new calculator works. GI college entrance exams such as the SAT. (A) -37 1 (B) - _ 4 (B) -25 Questions on the SAT represent the following math content areas: Number and Operations, 20–25% (D) 7 1 (D) _ 4 Algebra and Functions, 35–40% (E) 27 Geometry and Measurement, 25–30% (E) 1 5 2. Which of the following is equal to 4 ? Data Analysis, Statistics, and Probability, 10–15% 5. What is the area of a rectangle with a length of x - a and a width of x + b? ÝÊÊ> I. 3 5 × 1 5 (A) x 2 - a 2 II. 2 10 (B) x 2 + b 2 III. 4 0 × 4 5 (C) x 2 - abx + ab Items on this page focus on: ÝÊÊL (A) I only (D) x 2 - ax - bx - ab (B) II only (E) x 2 + bx - ax - ab • Algebra and Functions • Geometry and Measurement Text References: Item Lesson (C) I and II only (D) II and III only (E) I, II, and III Online Edition College Entrance Exam Practice (C) -5 (C) 0 @<I Resources 4. What is the value of 2x 3 - 4x 2 + 3x + 1 when x = -2? (A) -1 <D 6. For integers greater than 0, define the following operations. 1 7-8 2 7-1, 7-3 3 7-1 4 7-6 5 7-8 6 7-7 a □ b = 2a 2 + 3b 3. If x -4 = 81, then x = (A) -3 1 (B) _ 4 1 (C) _ 3 (D) 3 (E) 9 a b = 5a 2 - 2b What is (a □ b) + (a b)? (A) 7a 2 + b (B) -3a 2 + 5b (C) 7a 2 - b (D) 3a 2 - 5b (E) -3a 2 - b College Entrance Exam Practice 1. Students who chose A or E may have made a sign error in the second term. Suggest that students use parentheses around each product of binomials until the multiplication is complete and then distribute the negative sign from the subtraction of the terms. A1NL11S_c07_0530-0535.indd 535 2. Students who chose A, C, or E may have added the bases in choice I and kept the exponent the same. Remind students that if the bases are the same when powers are multiplied, then they can add the exponents. 3. Students who chose D found the value of x if x4 = 81. Suggest that students rewrite x-4 with a positive exponent and try again. 4. Students who chose C probably made a sign error in the first or second term. Remind students to be careful when using exponents with negative numbers. Be sure they understand when the negative sign is part of the base, and when it is not. 535 5. Students who did not choose E should review multiplying binomials. Remind 7/18/09 5:04:40 PM students to be cautious of signs. 6. Students who chose B subtracted the two binomials instead of adding them. Remind students to read each test item carefully. College Entrance Exam Practice 535 CHAPTER 7 Organizer Any Question Type: Use a Diagram Objective: Provide opportunities When a test item includes a diagram, use it to help solve the problem. Gather as much information from the drawing as possible. However, keep in mind that diagrams are not always drawn to scale and can be misleading. GI to learn and practice common testtaking strategies. <D @<I Online Edition Multiple Choice What is the height of the triangle when x = 4 and y = 1? This Test Tackler explains how test item diagrams can inadvertently mislead students. If students assume that visual information from a diagram is correct, they will likely misinterpret the information. Advise students that diagrams are not always drawn to scale. Students should not rely solely on the appearance of the drawing to answer the test item. Instead they should look closely at a drawing’s labels. They may even need to redraw the diagram to scale in order to better represent the problem. Explain that even though a test item may not include a diagram, it may be beneficial for students to make a quick sketch. Show students the importance of labeling their sketch with the information in the test item. 2 8 4 16 ÝÞ®Ó In the diagram, the height appears to be less than 6, so you might eliminate choices C and D. However, doing the math shows that the height is actually greater than 6. Do not rely solely on visual information. Always use the numbers given in the problem. È The height of the triangle is (xy)2. When x = 4 and y = 1, (xy) 2 = (4 · 1) 2 = (4) 2 = 16. Choice D is the correct answer. If a test item does not have a diagram, draw a quick sketch of the problem situation. Label your diagram with the data given in the problem. Short Response A square placemat is lying in the middle of a rectangular table. x The side length of the placemat is __ . The length of the table is 12x, and the 2 () width is 8x. Write a polynomial to represent the area of the placemat. Then write a polynomial to represent the area of the table that surrounds the placemat. £ÓÝ Use the information in the problem to draw and label a diagram. Then write the polynomials. area of placemat = s 2 = x (_2x ) = (_2x )(_2x ) = _ 4 2 area of table = w = (12x)(8x) = 96x 2 area of table - area of placemat = 96x 2 - Ý Ú Ó nÝ 2 Ú ÊÝÊ Ó x 384x - x = _ 383x _ =_ 2 2 4 4 2 2 4 2 x . The area of the placemat is __ 4 2 383x The area of the table that surrounds the placemat is _____ . 4 536 Chapter 7 Exponents and Polynomials A1NL11S_c07_0536-0537.indd Sec1:536 536 Chapter 7 6/25/09 9:52:13 AM If a given diagram does not reflect the problem, draw a sketch that is more accurate. If a test item does not have a diagram, use the given information to sketch your own. Try to make your sketch as accurate as possible. Answers Item C Short Response Write a polynomial expression 1. the width for the area of triangle QRP. Write a polynomial expression for the area of triangle MNP. Then use these expressions to write a polynomial expression for the area of QRNM. Read each test item and answer the questions that follow. 2. Possible answer: By drawing a rectangle and labeling each dimension, you will have a better understanding of what should be substituted into the area formula for length and width. + Item A Short Response The width of a rectangle is £ä 3. Ý 1.5 feet more than 4 times its length. Write a polynomial expression for the area of the rectangle. What is the area when the length is 16.75 feet? * £ÊÊÝ x , È x 7. Describe how redrawing the figure can help you better understand the information in the problem. 1. What is the unknown measure in this problem? 2. How will drawing a diagram help you solve the problem? 4. greater 5. No; it appears that the length of rectangle ABDC is less than the length of rectangle MNPO, and this is not consistent with the dimensions that are given. 8. After reading this test item, a student redrew the figure as shown below. Is this a correct interpretation of the original figure? Explain. 3. Draw and label a sketch of the situation. 6. Possible answer: A Item B Multiple Choice Rectangle ABDC is similar to rectangle MNPO. If the width of rectangle ABDC is 8, what is its length? D C £ÓÝ M { n B " * 2 2x 24x 24 4. Look at the dimensions in the diagram. Do you think that the length of rectangle ABDC is greater or less than the length of rectangle MNPO? 5. Do you think the drawings reflect the information in the problem accurately? Why or why not? Item D Multiple Choice The measure of angle XYZ O is (x 2 + 10x + 15)°. What is the measure of angle XYW ? (6x + 15)° (2x 2 + 14x + 15)° (6x 2 + 15)° 8 ÝÓÊÊ{Ý®Â 9 N < 8. No, it is not correct. The student mislabeled the base of triangle QRP. It should be labeled 7 + x. 9. What information does the diagram provide that the problem does not? 9. the measure of angle WYZ 10. Will the measure of angle XYW be less than or greater than the measure of angle XYZ ? Explain. 10. Less than; angle XYW lies within angle XYZ, so angle XYW must be smaller. 6. Draw your own sketch to match the information in the problem. Test Tackler P 7. Possible answer: By redrawing the figures into 2 separate triangles, you can better see the base and height measures, which will help you correctly set up the area formula for each triangle. 7 (14x + 15)° x 537 Answers to Test Items A. 4x2 + 1.5x; 1147.375 ft2 B. C A1NL11S_c07_0536-0537.indd Sec1:537 6/25/09 9:52:35 AM 9 1 1 C. 5x + 35; _(x2 + x); -_ x2 + _x + 35 2 2 2 D. F KEYWORD: MA7 Resources Test Tackler 537 CHAPTER 7 KEYWORD: MA7 TestPrep Organizer CUMULATIVE ASSESSMENT, CHAPTERS 1–7 Objective: Provide review GI and practice for Chapters 1–7 and standardized tests. <D @<I Multiple Choice 6. A restaurant claims to have served 352 × 10 6 hamburgers. What is this number in scientific notation? 1. A negative number is raised to a power. The result is a negative number. What do you know about the power? Online Edition Resources It is an even number. 3.52 × 10 8 It is an odd number. 3.52 × 10 4 It is zero. 352 × 10 6 It is a whole number. Assessment Resources 3.52 × 10 6 7. Janet is ordering game cartridges from an online retailer. The retailer’s prices, including shipping and handling, are given in the table below. 2. Which expression represents the phrase eight less than the product of a number and two? Chapter 7 Cumulative Test KEYWORD: MA7 TestPrep 2 - 8x Game Cartridges 8 - 2x 1 54.95 2x - 8 2 104.95 3 154.95 4 204.95 x -8 _ 2 3. An Internet service provider charges a $20 set-up fee plus $12 per month. A competitor charges $15 per month. Which equation can you use to find x, the number of months when the total charge will be the same for both companies? Total Cost ($) Which equation best describes the relationship between the total cost c and the number of game cartridges g? c = 54.95g 15 = 20 + 12x c = 51g + 0.95 20 + 12x = 15x c = 50g + 4.95 20x + 12 = 15x c = 51.65g 20 = 15x + 12x 8. Which equation describes a line parallel to y = 5 - 2x? 4. Which is a solution of the inequality 7 - 3(x - 3) > 2(x + 3)? y = -2x + 8 1x y=5+_ 2 y = 2x - 5 1x y=5-_ 2 0 2 5 12 9. A square has sides of length x - 4. A rectangle has a length of x + 2 and a width of 2x - 1. What is the total combined area of the square and the rectangle? 5. One dose of Ted’s medication contains 0.625 milligram, or _58_ milligram, of a drug. Which expression is equivalent to 0.625? 10x - 14 4x - 3 5(4) -2 3x 2 - 5x + 14 5(2) -4 3x 2 + 3x - 18 5(-2) 3 5(2) -3 538 Chapter 7 Exponents and Polynomials Answers 1. B 14. 2 2. H 15. 4 A11NLS_c07_0538-0539.indd 538 3. B 16. 16 4. F 17a. $28.00 5. D b. $19.60 6. G 7. C 8. F 9. C 10. J KEYWORD: MA7 Resources 538 Chapter 7 3 , or 0.75 13. __ 4 11. C 12. H 12/14/09 11:23:04 A A Test writers develop multiple-choice test options with distracters. Distracters are incorrect options that are based on common student errors. Be cautious! Even if the answer you calculated is one of the options, it may not be the correct answer. Always check your work carefully. 10. Jennifer has a pocketful of change, all in nickels and quarters. There are 11 coins with a total value of $1.15. Which system of equations can you use to find the number of each type of coin? ⎧ n + q = 11 ⎨ ⎩ n + q = 1.15 Short Response a. Find the price of the sweater while on the sale rack. Show your work. 1 Point = The student’s answer contains attributes of an appropriate response but is flawed. b. Find the price of the sweater while on the clearance rack. Show your work. 0 Points = The student’s answer contains no attributes of an appropriate response. a. Find a 2, b 2, and c 2 when a = 2x, b = x 2 - 1, and c = x 2 + 1. Show your work. b. Is (2x, x 2 - 1, x 2 + 1) a Pythagorean triple? Extended-Response Rubric Explain your reasoning. 19. Ron is making an ice sculpture. The block of ice is Item 21 in the shape of a rectangular prism with a length of (x + 2) inches, a width of (x - 2) inches, and a height of 2x inches. ⎧ n + q = 11 ⎨ ⎩ 0.05n + 0.25q = 1.15 4 Points = The student writes the correct expression with full work or explanation in part a, shows one method of checking the previous answer in part b, writes the correct area as a binomial square in part c, expands the product correctly and identifies the type of polynomial in part d, and answers correctly with explanation in part e. a. Write and simplify a polynomial expression for 11. Which of the following is a true statement? the volume of the block of ice. Show your work. p ⎡(a m) n ⎤ = a m+n+p ⎣ ⎦ p ⎡(a m) n ⎤ = a mn+p ⎣ ⎦ b. The final volume of the ice sculpture is (x 3 + 4x 2 - 10x + 1) cubic inches. Write an expression for the volume of ice that Ron carved away. Show your work. p ⎡(a m) n ⎤ = a mnp ⎣ ⎦ p ⎡(a m) n ⎤ = (a m+n) p ⎣ ⎦ 20. Simplify the expression (3 · a 2 · b -4 · a · b -3) 12. In 1867, the United States purchased the Alaska Territory from Russia for $7.2 × 10 6. The total area was about 6 × 10 5 square miles. What was the price per square mile? About $0.12 per square mile -3 using two different methods. Show that the results are the same. 3 Points = The student writes the correct expression with minimal work or explanation in part a, shows one method of checking the previous answer in part b, writes the area as a binomial square with minor errors in part c, expands the product with minor errors and identifies the type of polynomial in part d, and answers correctly with explanation in part e. Extended Response 21. Look at the pentagon below. About $1.20 per square mile About $12.00 per square mile 2x - 2 About $120.00 per square mile 4x - 4 Gridded Response 4x - 4 13. Evaluate the expression 3b-2c 0 for b = 2 and c = -3. a. Write and simplify an expression that represents the area of the pentagon. Show your work or explain your answer. 14. What is the slope of the line described by -3y = -6x - 12? 2 Points = The student answers parts a, b, and e correctly with attempted explanation; or the student answers parts c, d, and e correctly with attempted explanation. b. Show one method of checking that your 15. The quotient (5.6 × 10 ) ÷ (8 × 10 ) is written in scientific notation as (7 × 10 n). What is the value 8 expression in part a is correct. 3 c. The triangular part of the pentagon can be rearranged to form a square. Write the area of this square as the square of a binomial. of n? d. Expand the product that you wrote in part c. 16. The volume of a plastic cylinder is 64 cubic centimeters. A glass cylinder has the same height and a radius that is half that of the plastic cylinder. What is the volume in cubic centimeters of the glass cylinder? What type of polynomial is this? e. Is the square of a binomial ever a binomial? Explain your reasoning. Cumulative Assessment, Chapters 1–7 -3 20. (3 · a2 · b-4 · a · b-3) = Answers 18a. 2 Points = The student’s answer is an accurate and complete execution of the task or tasks. Pythagorean triple if a 2 + b 2 = c 2. ⎧ 5n + 25q = 11 ⎨ ⎩ n + q = 1.15 PM A1NL11S_c07_0538-0539.indd 539 Items 17–20 down 20% and placed on the sale rack. Later, the sweater was marked down an additional 30% and placed on the clearance rack. 18. A set of positive integers (a, b, c) is called a ⎧ n + q = 11 ⎨ ⎩ 5n + 25q = 1.15 a2 Short-Response Rubric 17. A sweater that normally sells for $35 was marked = (2x) = 2 4x2 = (x2 - 1)2 = x4 2x2 + 1 b2 c2 = (x2 + 1)2 = x4 + 2x2 + 1 b. Yes; (2x)2 + (x2 - 1)2 = 4x2 + x4 - 2x2 + 1 = x4 + 2x2 + 1 and (x2 + 1)2 = x4 + 2x2 + 1. Because (2x)2 + (x2 - 1)2 = (x2 + 1)2, the expressions form a Pythagorean triple. 19a. (2x3 - 8x) in3 b. (x3 - 4x2 + 2x - 1) in3 (3 · a3 · -3 b-7 ) = 3-3 · a-9 · b21 = b21 _ ; 27a9 (3 · a2 · b-4 · a · b-3)-3 = 3-3 · a-6 · b12 · a-3 · b9 = 3-3 · a-9 · b21 = b21 _ 27a9 21a. 20x2 - 40x + 20 b. Possible answer: Substitute 5 for x in the diagram. Then the side lengths of the square are 16 units, and the height of the triangle is 8 units. The area of the 539 square is 256 square units, and the area of the triangle is 64 square7/18/09 units. Add 256 to 64, which gives the area of the pentagon, 320 square units. Then substitute 5 for x in the simplified expression. It also simplifies to 320. 1 Point = The student answers one part correctly but does not attempt all parts; or the student attempts to answer all parts of the problem but does not correctly answer any part. 0 Points = The student does not answer correctly and does not attempt all parts of the problem. 5:05:34 PM c. (2x - 2)2 d. 4x2 - 8x + 4; trinomial e. No; for any values of a and b, (a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2. This is always a trinomial. Cumulative Assessment 539

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