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A CAREERS AND MATHEMATICS: Pharmacist Pharmacists distribute prescription drugs to individuals. They also advise patients, physicians, and other healthcare workers on the selection, dosages, interactions, and side effects of medications. They also monitor patients to ensure that they are using their medications safely and effectively. Some pharmacists specialize in oncology, nuclear pharmacy, geriatric pharmacy, or psychiatric pharmacy. © Istockphoto.com/Lucas Rucchin © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A Review of Basic Algebra Education and Mathematics Required • • A.1 Sets of Real Numbers A.2 Integer Exponents and Notation A.3 Rational Exponents and Radicals A.4 Polynomials A.5 Factoring Polynomials A.6 Rational Expressions Appendix Review Appendix Test Pharmacists are required to possess a Pharm.D. degree from an accredited college or school of pharmacy. This degree generally takes four years to complete. To be admitted to a Pharm.D. program, at least two years of college must be completed, which includes courses in the natural sciences, mathematics, humanities, and the social sciences. A series of examinations must also be passed to obtain a license to practice pharmacy. College Algebra, Trigonometry, Statistics, and Calculus I are courses required for admission to a Pharm.D. program. How Pharmacists Use Math and Who Employs Them • • Pharmacists use math throughout their work to calculate dosages of various drugs. These dosages are based on weight and whether the medication is given in pill form, by infusion, or intravenously. Most pharmacists work in a community setting, such as a retail drugstore, or in a healthcare facility, such as a hospital. Career Outlook and Salary • • Employment of pharmacists is expected to grow by 17 percent between 2008 and 2018, which is faster than the average for all occupations. Median annual wages of wage and salary pharmacists is approximately $106,410. In this appendix, we review many concepts and skills learned in previous algebra courses. For more information see: www.bls.gov/oco Not For Sale A1 11/16/11 10:08 AM A2 Appendix Not For Sale A Review of Basic Algebra A.1 Sets of Real Numbers In this section, we will learn to 5 3 7 6 1 9 5 9 8 6 8 6 4 8 3 7 2 6 2 8 4 1 9 8 7 SUDOKU 3 1 6 5 9 Sudoku, a game that involves number placement, is very popular. The objective is to fill a 9 by 9 grid so that each column, each row and each of the 3 by 3 blocks contains the numbers from 1 to 9. A partially completed Sudoku grid is shown in the margin. To solve Sudoku puzzles, logic and the set of numbers, 5 1, 2, 3, 4, 5, 6, 7, 8, 9 6 are used. Sets of numbers are important in mathematics, and we begin our study of algebra with this topic. A set is a collection of objects, such as a set of dishes or a set of golf clubs. The set of vowels in the English language can be denoted as 5 a, e, i, o, u 6 , where the braces 5 6 are read as “the set of.” If every member of one set B is also a member of another set A, we say that B is a subset of A. We can denote this by writing B ( A, where the symbol ( is read as “is a subset of.” (See Figure A.1 below.) If set B equals set A, we can write B 8 A. If A and B are two sets, we can form a new set consisting of all members that are in set A or set B or both. This set is called the union of A and B. We can denote this set by writing, A c B where the symbol c is read as “union.” (See Figure A.1 below.) We can also form the set consisting of all members that are in both set A and set B. This set is called the intersection of A and B. We can denote this set by writing A d B, where the symbol d is read as “intersection.” (See Figure A.1 below.) A B A BA B A AB B AB FIGURE A-1 EXAMPLE 1Understanding Subsets and Finding the Union and Intersection of Two Sets Let A 5 5 a, e, i 6 , B 5 5 c, d, e 6 , and V 5 5 a, e, i, o, u 6 . a. Is A ( V? b. Find A c B. c. Find A d B. SOLUTION a. Since each member of set A is also a member of set V, A ( V. b. The union of set A and set B contains the members of set A, set B, or both. Thus, A c B 5 5 a, c, d, e, i 6 . c. The intersection of set A and set B contains the members that are in both set A and set B. Thus, A d B 5 5 e 6 . Self Check 1 a. Is B ( V? b. Find B c V c. Find A d V Now Try Exercise 33. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 1.Identify sets of real numbers. 2.Identify properties of real numbers. 3.Graph subsets of real numbers on the number line. 4.Graph intervals on the number line. 5.Define absolute value. 6.Find distances on the number line. Section .1 Sets of Real Numbers A3 1. Identify Sets of Real Numbers There are several sets of numbers that we use in everyday life. Basic Sets of Numbers Natural numbers The numbers that we use for counting: 5 1, 2, 3, 4, 5, 6, % 6 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Whole numbers The set of natural numbers including 0: 5 0, 1, 2, 3, 4, 5, 6, % 6 Integers The set of whole numbers and their negatives: 5 % , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, % 6 In the definitions above, each group of three dots (called an ellipsis) indicates that the numbers continue forever in the indicated direction. Two important subsets of the natural numbers are the prime and composite numbers. A prime number is a natural number greater than 1 that is divisible only by itself and 1. A composite number is a natural number greater than 1 that is not prime. • The set of prime numbers: 5 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, % 6 • The set of composite numbers: 5 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, % 6 Two important subsets of the set of integers are the even and odd integers. The even integers are the integers that are exactly divisible by 2. The odd integers are the integers that are not exactly divisible by 2. • The set of even integers: 5 % , 210, 28, 26, 24, 22, 0, 2, 4, 6, 8, 10, % 6 • The set of odd integers: 5 % , 29, 27, 25, 23, 21, 1, 3, 5, 7, 9, % 6 So far, we have listed numbers inside braces to specify sets. This method is called the roster method. When we give a rule to determine which numbers are in a set, we are using set-builder notation. To use set-builder notation to denote the set of prime numbers, we write 5 x 0 x is a prime number 6 Read as “the set of all numbers x such that x is a prime ▲ ▲ Caution variable such that Remember that the denominator of a fraction can never be 0. ▲ number.” Recall that when a letter stands for a number, it is called a variable. rule that determines membership in the set The fractions of arithmetic are called rational numbers. Rational Numbers Rational numbers are fractions that have an integer numerator and a nonzero integer denominator. Using set-builder notation, the rational numbers are e a 0 a is an integer and b is a nonzero integer f b Rational numbers can be written as fractions or decimals. Some examples of rational numbers are 5 3 5 0.75, 5 5 , 1 4 1 5 5 20.333 % , 2 5 20.454545 % The = sign indicates 3 11 that two quantities Not For Sale 2 are equal. A4 Appendix Not For Sale A Review of Basic Algebra These examples suggest that the decimal forms of all rational numbers are either terminating decimals or repeating decimals. EXAMPLE 2Determining whether the Decimal Form of a Fraction Terminates or Repeats Determine whether the decimal form of each fraction terminates or repeats: a. 7 65 b. 16 99 7 7 a. To change 16 to a decimal, we perform a long division to get 16 5 0.4375. Since 7 0.4375 terminates, we can write 16 as a terminating decimal. 65 b. To change 65 99 to a decimal, we perform a long division to get 99 5 0.656565 % . 65 Since 0.656565 ... repeats, we can write 99 as a repeating decimal. We can write repeating decimals in compact form by using an overbar. For example, 0.656565 % 5 0.65. Self Check 2 Determine whether the decimal form of each fraction terminates or repeats: a. 38 99 7 b. 8 Now Try Exercise 35. Some numbers have decimal forms that neither terminate nor repeat. These nonterminating, nonrepeating decimals are called irrational numbers. Three examples of irrational numbers are 1.010010001000010 % "2 5 1.414213562 % and p 5 3.141592654 % The union of the set of rational numbers (the terminating and repeating decimals) and the set of irrational numbers (the nonterminating, nonrepeating decimals) is the set of real numbers (the set of all decimals). Real Numbers A real number is any number that is rational or irrational. Using set-builder notation, the set of real numbers is 5 x 0 x is a rational or an irrational number 6 EXAMPLE 3 Classifying Real Numbers In the set 5 23, 22, 0, 12, 1, "5, 2, 4, 5, 6 6 , list all a. even integers b. prime numbers c. rational numbers SOLUTION We will check to see whether each number is a member of the set of even integers, the set of prime numbers, and the set of rational numbers. a. even integers: –2, 0, 2, 4, 6 b. prime numbers: 2, 5 c. rational numbers: –3, –2, 0, 12, 1, 2, 4, 5, 6 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. SOLUTION In each case, we perform a long division and write the quotient as a decimal. Section .1 Sets of Real Numbers A5 Self Check 3 In the set in Example 3, list all a. odd integers b. composite numbers c. irrational numbers. Now Try Exercise 43. Figure A-2 shows how the previous sets of numbers are related. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Real numbers 8 11 −3, – – , −√2, 0, –– , π, 9.9 5 16 Rational numbers −6, –1.25, 0, 2– , 5 3– , 80 3 4 Noninteger rational numbers 111 – 13 ––, – 0.1, 2–, –––, 3.17 7 5 53 Irrational numbers −√5, π, √21, 2√101 Integers −4, −1, 0, 21 Negative integers −47, −17, −5, −1 Whole numbers 0, 1, 4, 8, 10, 53, 101 Zero 0 Natural numbers (positive integers) 1, 12, 38, 990 FIGURE A-2 2. Identify Properties of Real Numbers When we work with real numbers, we will use the following properties. Properties of Real Numbers If a, b, and c are real numbers, The Commutative Properties for Addition and Multiplication a 1 b 5 b 1 a ab 5 ba The Associative Properties for Addition and Multiplication 1a 1 b2 1 c 5 a 1 1b 1 c2 1ab2 c 5 a 1bc2 The Distributive Property of Multiplication over Addition or Subtraction a 1b 1 c2 5 ab 1 ac or a 1b 2 c2 5 ab 2 ac The Double Negative Rule 2 12a2 5 a Caution When the Associative Property is used, the order of the real numbers does not change. The real numbers that occur within the parentheses change. Not For Sale The Distributive Property also applies when more than two terms are within parentheses. Appendix A A6 Not For Sale A Review of Basic Algebra EXAMPLE 4 SOLUTION Identifying Properties of Real Numbers a. 19 1 22 1 3 5 9 1 12 1 32 b. 3 1x 1 y 1 22 5 3x 1 3y 1 3 ? 2 We will compare the form of each statement to the forms listed in the properties of real numbers box. a. This form matches the Associative Property of Addition. Self Check 4 a. mn 5 nm b. 1xy2 z 5 x 1yz2 c. p 1 q 5 q 1 p Now Try Exercise 17. 3. Graph Subsets of Real Numbers on the Number Line We can graph subsets of real numbers on the number line. The number line shown in Figure A-3 continues forever in both directions. The positive numbers are represented by the points to the right of 0, and the negative numbers are represented by the points to the left of 0. Negative numbers Comment –5 –4 –3 –2 Positive numbers –1 Zero is neither positive nor negative. 0 1 2 3 4 5 FIGURE A-3 Figure A-4(a) shows the graph of the natural numbers from 1 to 5. The point associated with each number is called the graph of the number, and the number is called the coordinate of its point. Figure A-4(b) shows the graph of the prime numbers that are less than 10. Figure A-4(c) shows the graph of the integers from 24 to 3. 7 3 Figure A-4(d) shows the graph of the real numbers 23, 24, 0.3, and "2. –1 Comment 0 1 2 3 4 5 6 0 1 2 3 4 (a) square with sides of length 1. –7/3 –5 –4 –3 –2 –1 2 1 6 7 8 9 10 (b) "2 can be shown as the diagonal of a 1 5 0 1 2 3 4 –3 –2 –3/4 –1 – 0.3 0 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. b. This form matches the Distributive Property. 2 1 2 (d) (c) FIGURE A-4 1 The graphs in Figure A-4 suggest that there is a one-to-one correspondence between the set of real numbers and the points on a number line. This means that to each real number there corresponds exactly one point on the number line, and to each point on the number line there corresponds exactly one real-number coordinate. 1 EXAMPLE 5 Graphing a Set of Numbers on a Number Line Graph the set U23, 243, 0, "5V. 11/16/11 10:08 AM Section .1 Sets of Real Numbers A7 SOLUTION We will mark (plot) each number on the number line. To the nearest tenth, "5 5 2.2. 5 – 4/3 –3 –2 –1 0 1 2 3 Self Check 5 Graph the set U22, 34, "3V. (Hint: To the nearest tenth, "3 5 1.7.) Now Try Exercise 51. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 4. Graph Intervals on the Number Line To show that two quantities are not equal, we can use an inequality symbol. Symbol Read as Examples 2 “is not equal to” 528 and 0.25 2 13 , “is less than” 12 , 20 and 0.17 , 1.1 . “is greater than” 15 . 9 and 1 2 # “is less than or equal to” 25 # 25 and 1.7 # 2.3 $ “is greater than or equal to” 19 $ 19 and 15.2 $ 13.7 < “is approximately equal to” "2 < 1.414 and "3 < 1.732 . 0.2 It is possible to write an inequality with the inequality symbol pointing in the opposite direction. For example, • 12 , 20 is equivalent to 20 . 12 • 2.3 $ 21.7 is equivalent to 21.7 # 2.3 In Figure A-3, the coordinates of points get larger as we move from left to right on a number line. Thus, if a and b are the coordinates of two points, the one to the right is the greater. This suggests the following facts: • If a . b, point a lies to the right of point b on a number line. • If a , b, point a lies to the left of point b on a number line. Figure A-5(a) shows the graph of the inequality x . 22 (or 22 , x). This graph includes all real numbers x that are greater than 22. The parenthesis at 22 indicates that 22 is not included in the graph. Figure A-5(b) shows the graph of x # 5 (or 5 $ x). The bracket at 5 indicates that 5 is included in the graph. ] ( −2 5 (a) (b) FIGURE A-5 Sometimes two inequalities can be written as a single expression called a compound inequality. For example, the compound inequality 5 , x , 12 is a combination of the inequalities 5 , x and x , 12. It is read as “5 is less than x, and x is less than 12,” and it means that x is between 5 and 12. Its graph is shown in Figure A-6. ( ) Not For Sale 5 12 FIGURE A-6 A8 Appendix Not For Sale A Review of Basic Algebra The graphs shown in Figures A-5 and A-6 are portions of a number line called intervals. The interval shown in Figure A-7(a) is denoted by the inequality 22 , x , 4, or in interval notation as 122, 42 . The parentheses indicate that the endpoints are not included. The interval shown in Figure A-7(b) is denoted by the inequality x . 1, or as 11, ` 2 in interval notation. ( ) 4 1 (1, ∞) x>1 x is greater than 1. (−2, 4) −2 < x < 4 x is between −2 and 4. (a) (b) FIGURE A-7 Caution The symbol ` (infinity) is not a real number. It is used to indicate that the graph in Figure A-7(b) extendsinfinitely far to right. A compound inequality such as 22 , x , 4 can be written as two separate inequalities: x . 22 and x , 4 This expression represents the intersection of two intervals. In interval notation, this expression can be written as 122, ` 2 d 12`, 42 Read the symbol d as “intersection.” Since the graph of 22 , x , 4 will include all points whose coordinates satisfy both x . 22 and x , 4 at the same time, its graph will include all points that are larger than 22 but less than 4. This is the interval 122, 42 , whose graph is shown in Figure A-7(a). EXAMPLE 6 Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality 23 , x , 5 in interval notation and graph it. SOLUTION This is the interval 123, 52 . Its graph includes all real numbers between 23 and 5, as shown in Figure A-8. ( −3 ) 5 FIGURE A-8 Self Check 6 Write the inequality 22 , x # 5 in interval notation and graph it. Now Try Exercise 63. If an interval extends forever in one direction, it is called an unbounded interval. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. ( −2 Section .1 Sets of Real Numbers A9 Unbounded Intervals Interval Inequality 1a, ` 2 x . a 12, ` 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 3 a, ` 2 ( ) −3 2 (−3, 2) −3 < x < 2 (a) [ ) −2 3 [−2, 3) −2 ≤ x < 3 (b) FIGURE A-9 [ −2 3 2, ` 2 12`, a2 x . 2 x $ 2 –2 –1 12`, 2 4 x # 2 ( 0 1 2 3 4 5 6 0 1 2 3 4 5 6 – 4 –3 –2 –1 0 1 3 4 0 1 3 4 [ a –2 –1 [ ( x , a x , 2 12`, ` 2 ( a x $ a 12`, 22 12`, a 4 Graph a ) 2 [ x # a a –4 –3 –2 –1 ] 2 2` , x , ` A bounded interval with no endpoints is called an open interval. Figure A-9(a) shows the open interval between 23 and 2. A bounded interval with one endpoint is called a half-open interval. Figure A-9(b) shows the half-open interval between 22 and 3, including 22. Intervals that include two endpoints are called closed intervals. Figure A-10 shows the graph of a closed interval from 22 to 4. ] 4 [−2, 4] −2 ≤ x ≤ 4 FIGURE A-10 Open Intervals, Half-Open Intervals, and Closed Intervals Interval Inequality Graph Open 1a, b2 a , x , b 122, 32 22 , x , 3 3 a, b2 a # x , b ( a ) ( b –4 –3 –2 –1 0 1 2 0 1 2 0 1 2 0 1 2 ) 3 4 5 4 5 4 5 4 5 Half-Open 3 22, 32 22 # x , 3 1a, b 4 a , x # b [ a ) [ b –4 –3 –2 –1 ) 3 Half-Open 122, 3 4 22 , x # 3 3 a, b 4 a # x # b ( a ] ( b –4 –3 –2 –1 ] 3 Closed 3 22, 3 4 22 # x # 3 [ a ] [ b –4 –3 –2 –1 Not For Sale ] 3 A10 Appendix Not For Sale A Review of Basic Algebra EXAMPLE 7 Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality 3 # x in interval notation and graph it. SOLUTION The inequality 3 # x can be written in the form x $ 3. This is the interval 3 3, ` 2 . Its graph includes all real numbers greater than or equal to 3, as shown in Figure A-11. [ 3 FIGURE A-11 Now Try Exercise 65. EXAMPLE 8 Writing an Inequality in Interval Notation and Graphing the Inequality Write the inequality 5 $ x $ 21 in interval notation and graph it. SOLUTION The inequality 5 $ x $ 21 can be written in the form 21 # x # 5 This is the interval 3 21, 5 4 . Its graph includes all real numbers from 21 to 5. The graph is shown in Figure A-12. [ ] −1 5 FIGURE A-12 Self Check 8 Write the inequality 0 # x # 3 in interval notation and graph it. Now Try Exercise 69. The expression x , 22 or x $ 3 Read as “x is less than 22 or x is greater than or equal to 3.” represents the union of two intervals. In interval notation, it is written as 12`, 222 c 3 3, ` 2 Read the symbol c as “union.” Its graph is shown in Figure A-13. ) [ −2 3 FIGURE A-13 5. Define Absolute Value The absolute value of a real number x (denoted as 0 x 0 ) is the distance on a number line between 0 and the point with a coordinate of x. For example, points with coordinates of 4 and 24 both lie four units from 0, as shown in Figure A-14. Therefore, it follows that 0 24 0 5 0 4 0 5 4 4 units –5 –4 –3 –2 –1 4 units 0 1 FIGURE A-14 2 3 4 5 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Self Check 7 Write the inequality 5 . x in interval notation and graph it. Section .1 Sets of Real Numbers A11 In general, for any real number x, 0 2x 0 5 0 x 0 We can define absolute value algebraically as follows. Absolute Value If x is a real number, then 0 x 0 5 x when x $ 0 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 0 x 0 5 2x when x , 0 Caution Remember that x is not always positive and 2x is not always negative. This definition indicates that when x is positive or 0, then x is its own absolute value. However, when x is negative, then 2x (which is positive) is its absolute value. Thus, 0 x 0 is always nonnegative. 0 x 0 $ 0 for all real numbers x EXAMPLE 9 Using the Definition of Absolute Value Write each number without using absolute value symbols: a. 0 3 0 b. 0 24 0 c. 0 0 0 d. 2 0 28 0 SOLUTION In each case, we will use the definition of absolute value. a. 0 3 0 5 3 b. 0 24 0 5 4 c. 0 0 0 5 0 d. 2 0 28 0 5 2 182 5 28 Self Check 9 Write each number without using absolute value symbols: a. 0 210 0 b. 0 12 0 c. 2 0 6 0 Now Try Exercise 85. In Example 10, we must determine whether the number inside the absolute value is positive or negative. EXAMPLE 10Simplifying an Expression with Absolute Value Symbols Write each number without using absolute value symbols: a. 0 p 2 1 0 b. 0 2 2 p 0 c. 0 2 2 x 0 if x $ 5 SOLUTION a. Since p < 3.1416, p 2 1 is positive, and p 2 1 is its own absolute value. 0p 2 10 5 p 2 1 b. Since 2 2 p is negative, its absolute value is 2 12 2 p2 . 0 2 2 p 0 5 2 12 2 p2 5 22 2 1 2 p2 5 22 1 p 5 p 2 2 c. Since x $ 5, the expression 2 2 x is negative, and its absolute value is 2 12 2 x2 . 0 2 2 x 0 5 2 12 2 x2 5 22 1 x 5 x 2 2 provided x $ 5 Self Check 10 Write each number without using absolute value symbols. 1Hint: "5 < 2.236.2 Not For Sale a. P 2 2 "5 P Now Try Exercise 89. b. 0 2 2 x 0 if x # 1 A12 Appendix Not For Sale A Review of Basic Algebra 6. Find Distances on the Number Line On the number line shown in Figure A-15, the distance between the points with coordinates of 1 and 4 is 4 2 1, or 3 units. However, if the subtraction were done in the other order, the result would be 1 2 4, or 23 units. To guarantee that the distance between two points is always positive, we can use absolute value symbols. Thus, the distance d between two points with coordinates of 1 and 4 is d 5 04 2 10 5 01 2 40 5 3 0 1 2 3 4 5 FIGURE A-15 In general, we have the following definition for the distance between two points on the number line. Distance between Two Points If a and b are the coordinates of two points on the number line, the distance between the points is given by the formula d 5 0b 2 a0 EXAMPLE 11 Finding the Distance between Two Points on a Number Line Find the distance on a number line between points with coordinates of a. 3 and 5 b. 22 and 3 c. 25 and 21 SOLUTION We will use the formula for finding the distance between two points. a. d 5 0 5 2 3 0 5 0 2 0 5 2 b. d 5 0 3 2 1222 0 5 0 3 1 2 0 5 0 5 0 5 5 c. d 5 0 21 2 1252 0 5 0 21 1 5 0 5 0 4 0 5 4 Self Check 11 Find the distance on a number line between points with coordinates of a. 4 and 10 b. 22 and 27 Now Try Exercise 99. Self Check Answers1. a. no b. {a, c, d, e, i, o, u} c. {a, e, i} 2. a. repeats b. terminates 3. a. 23, 1, 5 b. 4, 6 c. "5 4. a. Commutative Property of Multiplication b. Associative Property of Multiplication 3 –2 3/4 c. Commutative Property of Addition 5. –3 6. 122, 5 4 8. 3 0, 3 4 ( ] −2 [ 0 5 ] 7. 12`, 52 –2 –1 ) 5 9. a. 10 b. 12 c. 26 3 10. a. "5 2 2 b. 2 2 x 11. a. 6 b. 5 0 1 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. –1 d = |4 − 1| = 3 Section .1 Sets of Real Numbers A13 Exercises A.1 Getting Ready © Cengage Learning. All rights reserved. No distribution allowed without express authorization. You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. A is a collection of objects. 2. If every member of one set B is also a member of a second set A, then B is called a of A. 3. If A and B are two sets, the set that contains all members that are in sets A and B or both is called the of A and B. 4. If A and B are two sets, the set that contains all members that are in both sets is called the of A and B. 5. A real number is any number that can be expressed as a . 6. A is a letter that is used to represent a number. 7. The smallest prime number is . 8. All integers that are exactly divisible by 2 are called integers. 9. Natural numbers greater than 1 that are not prime are called numbers. 2 8 10. Fractions such as 3 , 2 , and 279 are called numbers. 11. Irrational numbers are that don’t terminate and don’t repeat. 12. The symbol is read as “is less than or equal to.” 13. On a number line, the numbers are to the left of 0. 14. The only integer that is neither positive nor negative is . 15. The Associative Property of Addition states that 1x 1 y2 1 z 5 . 16. The Commutative Property of Multiplication states that xy 5 . 17. Use the Distributive Property to complete the state. ment: 5 1m 1 22 5 18. The statement 1m 1 n2 p 5 p 1m 1 n2 illustrates the Property of . 19. The graph of an is a portion of a number line. 20. The graph of an open interval has endpoints. 21. The graph of a closed interval has endpoints. 22. The graph of a interval has one endpoint. 23. Except for 0, the absolute value of every number is . 24. The between two distinct points on a number line is always positive. Let N 5 the set of natural numbers W 5 the set of whole numbers Z 5 the set of integers Q 5 the set of rational numbers R 5 the set of real numbers Determine whether each statement is true or false. Read the symbol ( as “is a subset of.” 25. N ( W 26. Q ( R 27. Q ( N 28. Z ( Q 29. W ( Z 30. R ( Z Practice Let A 5 5 a,b,c,d,e 6 , B 5 1d,e,f,g 6 , and C 5 5 a,c,e,f 6 . Find each set. 31. A c B 33. A d C 32. A d B 34. B c C Determine whether the decimal form of each fraction terminates or repeats. 9 3 35. 36. 16 8 3 5 37. 38. 11 12 Consider the following set: 5 25,24,223,0,1,"2,2,2.75,6,7 6 . 39. Which numbers are natural numbers? 40. Which numbers are whole numbers? 41. Which numbers are integers? 42. Which numbers are rational numbers? 4 3. Which numbers are irrational numbers? 44. Which numbers are prime numbers? 45. Which numbers are composite numbers? 46. Which numbers are even integers? 47. Which numbers are odd integers? 48. Which numbers are negative numbers? Graph each subset of the real numbers on a number line. 49. The natural numbers between 1 and 5 Not For Sale Not For Sale Appendix A Review of Basic Algebra 50. The composite numbers less than 10 51. The prime numbers between 10 and 20 52. The integers from –2 to 4 53. The integers between –5 and 0 54. The even integers between –9 and –1 55. The odd integers between –6 and 4 56. 20.7, 1.75, and 378 Write each inequality in interval notation and graph the interval. 57. x . 2 58. x , 4 59. 0 , x , 5 61. x . 24 67. 25 , x # 0 60. 22 , x , 3 62. x , 3 63. 22 # x , 2 65. x # 5 64. 24 , x # 1 66. x $ 21 68. 23 # x , 4 69. 22 # x # 3 70. 24 # x # 4 71. 6 $ x $ 2 72. 3 $ x $ 22 Write each pair of inequalities as the intersection of two intervals and graph the result. 73. x . 25 and x , 4 Write each inequality as the union of two intervals and graph the result. 77. x , 22 or x . 2 78. x # 25 or x . 0 79. x # 21 or x $ 3 80. x , 23 or x $ 2 Write each expression without using absolute value symbols. 82. 0 217 0 81. 0 13 0 83. 0 0 0 84. 2 0 63 0 86. 0 225 0 85. 2 0 28 0 87. 2 0 32 0 88. 89. 0 p 2 5 0 90. 91. 0 p 2 p 0 92. 93. 0 x 1 1 0 and x $ 2 94. 2 0 26 0 08 2 p0 0 2p 0 0 x 1 1 0 and x # 2 2 95. 0 x 2 4 0 and x , 0 96. 0 x 2 7 0 and x . 10 Find the distance between each pair of points on the number line. 97. 3 and 8 98. 25 and 12 99. 28 and 23 100. 6 and 220 Applications 101. What subset of the real numbers would you use to describe the populations of several cities? 102. What subset of the real numbers would you use to describe the subdivisions of an inch on a ruler? 103. What subset of the real numbers would you use to report temperatures in several cities? 104. What subset of the real numbers would you use to describe the financial condition of a business? 74. x $ 23 and x , 6 75. x $ 28 and x # 23 76. x . 1 and x # 7 Discovery and Writing 105. Explain why 2x could be positive. 106. Explain why every integer is a rational number. 107. Is the statement 0 ab 0 5 0 a 0 ? 0 b 0 always true? Explain. 0a0 a 1b 2 02 always true? 108. Is the statement ` ` 5 0b0 b Explain. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A14 Section 109. Is the statement 0 a 1 b 0 5 0 a 0 1 0 b 0 always true? Explain. 111. Explain why it is incorrect to write a , b . c if a , b and b . c. 110. Under what conditions will the statement given in Exercise 109 be true? 112. Explain why 0 b 2 a 0 5 0 a 2 b 0 . A.2 Integer Exponents and Scientific Notation In this section, we will learn to 1.Define natural-number exponents. 2.Apply the rules of exponents. 3.Apply the rules for order of operations to evaluate expressions. 4.Express numbers in scientific notation. 5.Use scientific notation to simplify computations. bioraven/Shutterstock.com The number of cells in the human body is approximated to be one hundred trillion or 100,000,000,000,000. One hundred trillion is 1102 1102 1102 # # # 1102 , where ten occurs fourteen times. Fourteen factors of ten can be written as 1014. In this section, we will use integer exponents to represent repeated multiplication of numbers. 1. Define Natural-Number Exponents When two or more quantities are multiplied together, each quantity is called a factor of the product. The exponential expression x4 indicates that x is to be used as a factor four times. 4 factors of x x4 5 x ? x ? x ? x In general, the following is true. Natural-Number Exponents For any natural number n, n factors of x xn 5 x ? x ? x ? c ? x © Cengage Learning. All rights reserved. No distribution allowed without express authorization. .2 Integer Exponents and Scientific Notation A15 In the exponential expression xn, x is called the base, and n is called the exponent or the power to which the base is raised. The expression xn is called a power of x. From the definition, we see that a natural-number exponent indicates how many times the base of an exponential expression is to be used as a factor in a product. If an exponent is 1, the 1 is usually not written: x1 5 x EXAMPLE 1 Using the Definition of Natural-Number Exponents Not For Sale Write each expression without using exponents: a. 42 b. 1242 2 c. 253 d. 1252 3 e. 3x4 f. 13x2 4 A16 Appendix Not For Sale A Review of Basic Algebra SOLUTION In each case, we apply the definition of natural-number exponents. a. 42 5 4 ? 4 5 16 Read 42 as “four squared.” b. 1242 2 5 1242 1242 5 16 c. 253 5 25 152 152 5 2125 Read 1242 2 as “negative four squared.” Read 253 as “the negative of five cubed.” d. 1252 3 5 1252 1252 1252 5 2125 e. 3x4 5 3 ? x ? x ? x ? x Read 1252 3 as “negative five cubed.” Read 3x4 as “3 times x to the fourth power.” Self Check 1 Write each expression without using exponents: a. 73 b. 1232 2 c. 5a3 d. 15a2 4 Now Try Exercise 19. Comment n factors of x axn 5 a ? x ? x ? x ? c ? x n factors of ax 1ax2 n 5 1ax2 1ax2 1ax2 ? c ? 1ax2 Note the distinction between axn and 1ax2 n: Also note the distinction between 2xn and 12x2 n: Accent on Technology n factors of 2x 12x2 n 5 12x2 12x2 12x2 ? c ? 12x2 n factors of x 2xn 5 2 1x ? x ? x ? c ? x2 Using a Calculator to Find Powers We can use a graphing calculator to find powers of numbers. For example, consider 2.353. • Input 2.35 and press the • Input 3 and press ENTER . key. Comment To find powers on a scientific calculator use the yx key. Figure A-16 We see that 2.353 5 12.977875 as the figure shows above. 2. Apply the Rules of Exponents We begin the review of the rules of exponents by considering the product xmxn. Since xm indicates that x is to be used as a factor m times, and since xn indicates that x is to be used as a factor n times, there are m 1 n factors of x in the product xmxn. m 1 n factors of x m factors of x n factors of x x x 5 x ? x ? x ? c ? x ? x ? x ? x ? c ? x 5 xm1n m n © Cengage Learning. All rights reserved. No distribution allowed without express authorization. f. 13x2 4 5 13x2 13x2 13x2 13x2 5 81 ? x ? x ? x ? x Read 13x2 4 as “3x to the fourth power.” Section .2 Integer Exponents and Scientific Notation A17 This suggests that to multiply exponential expressions with the same base, we keep the base and add the exponents. Product Rule for Exponents If m and n are natural numbers, then xmxn 5 xm1n mn factors of x The Product Rule applies to exponential expressions with the same base. A product of two powers with different bases, such as x4y3, cannot be simplified. To find another property of exponents, we consider the exponential expression 1xm2 n. In this expression, the exponent n indicates that xm is to be used as a factor n times. This implies that x is to be used as a factor mn times. n factors of xm 1x 2 5 1x 2 1xm2 1xm2 ? c ? 1xm2 5 xmn m n m This suggests that to raise an exponential expression to a power, we keep the base and multiply the exponents. To raise a product to a power, we raise each factor to that power. n factors of xy n factors of x n factors of y 1xy2 n 5 1xy2 1xy2 1xy2 ? c ? 1xy2 5 1x ? x ? x ? c ? x2 1y ? y ? y ? c ? y2 5 xnyn To raise a fraction to a power, we raise both the numerator and the denominator to that power. If y 2 0, then n factors of x y x n x x x x a b 5 a ba ba b ? c ? a b y y y y y n factors of x xxx ? c ? x yyy ? c ? y 5 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Comment n factors of y 5 xn yn The previous three results are called the Power Rules of Exponents. Power Rules of Exponents If m and n are natural numbers, then x n xn 1xm2 n 5 xmn 1xy2 n 5 xnyn a b 5 n 1y 2 02 y y EXAMPLE 2 Using Exponent Rules to Simplify Expressions with Natural-Number Exponents Simplify: a. x5x7 b. x2y3x5y c. 1x42 9 d. 1x2x52 3 Not For Sale e. a 5x2y 2 x 5 b 2 b f. a z3 y A18 Appendix Not For Sale A Review of Basic Algebra SOLUTION In each case, we will apply the appropriate rule of exponents. a. x5x7 5 x517 5 x12 b. x2y3x5y 5 x215y311 5 x7y4 c. 1x42 9 5 x4?9 5 x36 d. 1x2x52 3 5 1x7 2 3 5 x21 f. a x5 x5 x 5 1y 2 02 2b 5 2 5 5 10 1y 2 y y 5x2y 2 52 1x22 2y2 25x4y2 5 5 1z 2 02 b 1z32 2 z3 z6 Self Check 2 Simplify: a. 1y32 2 b. 1a2a42 3 Now Try Exercise 49. c. 1x22 3 1x32 2 d. a 3a3b2 3 b 1c 2 02 c3 If we assume that the rules for natural-number exponents hold for exponents of 0, we can write x0xn 5 x01n 5 xn 5 1xn Since x0xn 5 1xn, it follows that if x 2 0, then x0 5 1. Zero Exponent x0 5 1 1x 2 02 If we assume that the rules for natural-number exponents hold for exponents that are negative integers, we can write x2nxn 5 x2n1n 5 x0 5 1 1x 2 02 However, we know that 1 ? xn 5 1 1x 2 02 xn xn 1 n n ? x 5 n , and any nonzero number divided by itself is 1. x x Since x2nxn 5 x1n ? xn , it follows that x2n 5 x1n 1x 2 02 . Negative Exponents If n is an integer and x 2 0, then x2n 5 1 1 5 xn n and x x2n Because of the previous definitions, all of the rules for natural-number exponents will hold for integer exponents. EXAMPLE 3Simplifying Expressions with Integer Exponents Simplify and write all answers without using negative exponents: 1 a. 13x2 0 b. 3 1x02 c. x24 d. 26 e. x23x f. 1x24x82 25 x SOLUTION We will use the definitions of zero exponent and negative exponents to simplify each expression. 1 1 a. 13x2 0 5 1 b. 3 1x02 5 3 112 5 3 c. x24 5 4 d. 26 5 x6 x x © Cengage Learning. All rights reserved. No distribution allowed without express authorization. e. a Section .2 Integer Exponents and Scientific Notation A19 e. x23x 5 x2311 f. 1x24x82 25 5 1x42 25 5 x22 5 5 x220 1 x2 5 1 x20 Self Check 3 Simplify and write all answers without using negative exponents: a. 7a0 b. 3a22 c. a24a2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Now Try Exercise 59. d. 1a3a272 3 To develop the Quotient Rule for Exponents, we proceed as follows: xm m 1 m 2n 5 xm1 1 2n2 5 xm2n 1x 2 02 n 5 x a nb 5 x x x x This suggests that to divide two exponential expressions with the same nonzero base, we keep the base and subtract the exponent in the denominator from the exponent in the numerator. Quotient Rule for Exponents If m and n are integers, then xm 5 xm2n 1x 2 02 xn EXAMPLE 4Simplifying Expressions with Integer Exponents Simplify and write all answers without using negative exponents: a. x8 x2x4 b. x5 x25 SOLUTION We will apply the Product and Quotient Rules of Exponents. a. x8 x2x4 x6 825 5 x b. 5 x5 x25 x25 5 x62 1252 5 x3 5 x11 Self Check 4 Simplify and write all answers without using negative exponents: a. x26 x2 b. x4x23 x2 Now Try Exercise 69. EXAMPLE 5Simplifying Expressions with Integer Exponents Simplify and write all answers without using negative exponents: Not For Sale a. a x3y22 22 x 2n b. b a b x22y3 y A20 Appendix Not For Sale A Review of Basic Algebra SOLUTION We will apply the appropriate rules of exponents. x3y22 22 1 2 b 5 1x32 22 y22232 22 x22y3 5 1x5y252 22 5 x210y10 5 y10 x10 x 2n x2n b. a b 5 2n y y 5 x2nxnyn xnyn 2n n n Multiply numerator and denominator by 1 in the form xnyn . y xy 5 x0yn y0xn 5 yn xn y n 5a b x x2nxn 5 x0 and y2nyn 5 y0. x0 5 1 and y0 5 1. Self Check 5 Simplify and write all answers without using negative exponents: a. a x4y23 2 b x23y2 b. a Now Try Exercise 75. 2a 23 b 3b Part b of Example 5 establishes the following rule. A Fraction to a Negative Power If n is a natural number, then x 2n y n a b 5 a b (x 2 0 and y 2 0) y x 3. Apply the Rules for Order of Operations to Evaluate Expressions When several operations occur in an expression, we must perform the operations in the following order to get the correct result. Strategy for Evaluating Expressions Using Order of Operations If an expression does not contain grouping symbols such as parentheses or brackets, follow these steps: 1. Find the values of any exponential expressions. 2. Perform all multiplications and/or divisions, working from left to right. 3. Perform all additions and/or subtractions, working from left to right. • If an expression contains grouping symbols such as parentheses, brackets, or braces, use the rules above to perform the calculations within each pair of grouping symbols, working from the innermost pair to the outermost pair. • In a fraction, simplify the numerator and the denominator of the fraction separately. Then simplify the fraction, if possible. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. a. a Section Comment .2 Integer Exponents and Scientific Notation A21 Many students remember the Order of Operations Rule with the acronym PEMDAS: • Parentheses • Exponents • Multiplication • Division • Addition • Subtraction © Cengage Learning. All rights reserved. No distribution allowed without express authorization. For example, to simplify 3 3 4 2 16 1 102 4 , we proceed as follows: 22 2 16 1 72 3 3 4 2 16 1 102 4 3 14 2 162 5 2 22 2 16 1 72 2 2 16 1 72 3 12122 5 2 2 2 13 3 12122 5 4 2 13 236 5 29 54 Simplify within the inner parentheses: 6 1 10 5 16. Simplify within the parentheses: 4 2 16 5 212, and 6 1 7 5 13. Evaluate the power: 22 5 4. 3 12122 5 236. 4 2 13 5 29. EXAMPLE 6Evaluating Algebraic Expressions If x 5 22, y 5 3, and z 5 24, evaluate a. 2x2 1 y2z b. 2z3 2 3y2 5x2 SOLUTION In each part, we will substitute the numbers for the variables, apply the rules of order of operations, and simplify. a. 2x2 1 y2z 5 2 1222 2 1 32 1242 Evaluate the powers. 5 2 142 1 9 1242 Do the multiplication. 5 240 Do the addition. b. 5 24 1 12362 2z3 2 3y2 2 1242 3 2 3 132 2 5 2 5x 5 1222 2 2 12642 2 3 192 5 5 142 2128 2 27 5 20 2155 5 20 31 52 4 Self Check 6 If x 5 3 and y 5 22, evaluate Evaluate the powers. Do the multiplications. Do the subtraction. Simplify the fraction. 2x2 2 3y2 . x2y Now Try Exercise 91. 4. Express Numbers in Scientific Notation Not For Sale Scientists often work with numbers that are very large or very small. These numbers can be written compactly by expressing them in scientific notation. A22 Appendix Not For Sale A Review of Basic Algebra Scientific Notation A number is written in scientific notation when it is written in the form N 3 10n Light travels 29,980,000,000 centimeters per second. To express this number in scientific notation, we must write it as the product of a number between 1 and 10 and some integer power of 10. The number 2.998 lies between 1 and 10. To get 29,980,000,000, the decimal point in 2.998 must be moved ten places to the right. This is accomplished by multiplying 2.998 by 1010. 29,980,000,000 5 2.998 3 1010 Scientific notation ▲ ▲ Standard notation ▲ Standard notation 0.0006214 5 6214 3 1024 Scientific notation ▲ ollirg/Shutterstock.com One meter is approximately 0.0006214 mile. To express this number in scientific notation, we must write it as the product of a number between 1 and 10 and some integer power of 10. The number 6.214 lies between 1 and 10. To get 0.0006214, the decimal point in 6.214 must be moved four places to the left. This is accomplished by multiplying 6.214 by 101 4 or by multiplying 6.214 by 1024. To write each of the following numbers in scientific notation, we start to the right of the first nonzero digit and count to the decimal point. The exponent gives the number of places the decimal point moves, and the sign of the exponent indicates the direction in which it moves. a. 3 7 2 0 0 0 5 3.72 3 105 5 places to the right. b. 0 . 0 0 0 5 3 7 5 5.37 3 1024 4 places to the left. c. 7.36 5 7.36 3 100 No movement of the decimal point. EXAMPLE 7 Writing Numbers in Scientific Notation Write each number in scientific notation: a. 62,000 b. 20.0027 SOLUTION a. We must express 62,000 as a product of a number between 1 and 10 and some integer power of 10. This is accomplished by multiplying 6.2 by 104. 62,000 5 6.2 3 104 b. We must express 20.0027 as a product of a number whose absolute value is between 1 and 10 and some integer power of 10. This is accomplished by multiplying 22.7 by 1023. 20.0027 5 22.7 3 1023 Self Check 7 Write each number in scientific notation: b. 0.0000087 a. 293,000,000 Now Try Exercise 103. EXAMPLE 8 Writing Numbers in Standard Notation Write each number in standard notation: a. 7.35 3 102 b. 3.27 3 1025 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. where 1 # 0 N 0 , 10 and n is an integer. Section .2 Integer Exponents and Scientific Notation A23 SOLUTION a. The factor of 102 indicates that 7.35 must be multiplied by 2 factors of 10. Because each multiplication by 10 moves the decimal point one place to the right, we have 7.35 3 102 5 735 b. The factor of 1025 indicates that 3.27 must be divided by 5 factors of 10. Because each division by 10 moves the decimal point one place to the left, we have 3.27 3 1025 5 0.0000327 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Self Check 8 Write each number in standard notation: a. 6.3 3 103 b. 9.1 3 1024 Now Try Exercise 111. 5. Use Scientific Notation to Simplify Computations Another advantage of scientific notation becomes evident when we multiply and divide very large and very small numbers. EXAMPLE 9 Using Scientific Notation to Simplify Computations Use scientific notation to calculate 13,400,0002 10.000022 . 170,000,000 SOLUTION After changing each number to scientific notation, we can do the arithmetic on the numbers and the exponential expressions separately. 13,400,0002 10.000022 13.4 3 1062 12.0 3 10252 5 170,000,000 1.7 3 108 6.8 5 3 1061 1 252 28 1.7 5 4.0 3 1027 5 0.0000004 Self Check 9 Use scientific notation to simplify Now Try Exercise 119. Accent on Technology 1192,0002 10.00152 . 10.00322 14,5002 Scientific Notation Graphing calculators will often give an answer in scientific notation. For example, consider 218. Figure A-17 Not For Sale We see in the figure that the answer given is 3.782285936E10, which means 3.782285936 3 1010. Appendix A A24 Not For Sale A Review of Basic Algebra Comment 218 on a scientific calculator, it is necessary to convert 0.000000000061 the denominator to scientific notation because the number has too many digits to fit on the screen. To calculate an expression like • For scientific notation, we enter these numbers and press these keys: 6.1 EXP 11 1/2 . • To evaluate the expression above, we enter these numbers and press these keys: 21 yx 8 5 4 6.1 EXP 11 1/2 5 Self Check Answers 1. a. 7 ? 7 ? 7 5 343 b. 1232 1232 5 9 c. 5 ? a ? a ? a d. 15a2 15a2 15a2 15a2 5 625 ? a ? a ? a ? a 2. a. y6 b. a18 c. x12 27a9b6 3 1 1 1 1 3. a. 7 b. 2 c. 2 d. 12 4. a. 8 b. 9 c a a a x x x14 27b3 6 5. a. 10 b. 6. 7. a. 29.3 3 107 b. 8.7 3 1026 y 8a3 5 8. a. 6,300 b. 0.00091 9. 20 d. Exercises A.2 Getting Ready 15. 252 You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. Each quantity in a product is called a of the product. 2. A number exponent tells how many times a base is used as a factor. 3. In the expression 12x2 3 , is the exponent and is the base. 4. The expression xn is called an expression. 5. A number is in notation when it is written in the form N 3 10n , where 1 # 0 N 0 , 10 and n is an . 6. Unless indicate otherwise, are performed before additions. Complete each exponent rule. Assume x 2 0. 7. xmxn 5 8. 1xm2 n 5 9. 1xy2 n 5 11. x0 5 10. xm 5 xn 12. x2n 5 Practice Write each number or expression without using exponents. 13. 132 14. 103 16. 1252 2 18. 14x2 3 1 7. 4x3 19. 125x2 4 21. 28x4 20. 26x2 22. 128x2 4 Write each expression using exponents. 23. 7xxx 25. 12x2 12x2 27. 13t2 13t2 123t2 29. xxxyy 24. 28yyyy 26. 12a2 12a2 12a2 28. 2 12b2 12b2 12b2 12b2 30. aaabbbb Use a calculator to simplify each expression. 31. 2.23 33. 20.54 32. 7.14 34. 120.22 4 Simplify each expression. Write all answers without using negative exponents. Assume that all variables are restricted to those numbers for which the expression is defined. 35. x2x3 37. 1z22 3 39. 1y y 2 2 3 4 5 41. 1z 2 1z 2 2 3 4 2 43. 1a 2 1a 2 5 2 3 45. 13x2 3 47. 1x2y2 3 36. y3y4 38. 1t62 7 40. 1a3a62 a4 42. 1t32 4 1t52 2 44. 1a22 4 1a32 3 46. 122y2 4 48. 1x3z42 6 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. The display will read 6.20046874820 . In standard notation, the answer is approximately 620,046,874,800,000,000,000. Section 49. a a2 3 b b 50. a 51. 12x2 0 53. 14x2 0 55. z24 57. y y 54. 22x0 1 56. 22 t 58. 2m22m3 59. 1x x 2 3 24 22 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 61. 63. 65. 67. 69. 71. 75. a 77. a 79. a 82. r5 62. 2 r t13 64. 4 t s9s3 66. 2 2 1s 2 t4 3 68. a 3 b t r9r23 70. 22 3 1r 2 x x3 a21 a17 1x22 2 x2x 3 3 m a 2b n 1a32 22 aa2 a23 24 a 21 b b 73. a 81. 60. 1y y 2 72. a 2 rr b r3r23 x5y22 4 b x23y2 74. 22 78. a 3x5y23 22 b 6x25y3 80. a 1822z23y2 21 15y2z222 3 15yz222 21 t24 22 b t23 Simplify each expression. 5 3 62 1 19 2 52 4 83. 2 4 12 2 32 2 95. 23x23z22 6x2z23 102. 0.00052 103. 20.000000693 104. 20.000000089 105. one trillion 106. one millionth x27y5 3 b x7y24 2 25 109. 2.21 3 1025 110. 2.774 3 1022 111. 0.00032 3 104 112. 9,300.0 3 1024 113. 23.2 3 1023 114. 27.25 3 103 116. 3x y b 2x22y26 12x24y3z25 3 b 4x4y23z5 117. 118. 119. 120. 165,0002 145,0002 250,000 10.0000000452 10.000000122 45,000,000 10.000000352 1170,0002 0.00000085 10.00000001442 112,0002 600,000 145,000,000,0002 1212,0002 0.00018 10.000000002752 14,7502 500,000,000,000 Applications 84. 6 3 3 2 14 2 72 2 4 25 12 2 422 3 108. 4.26 3 109 23 88. 2x3 107. 9.37 3 105 115. Let x 5 22, y 5 0, and z 5 3 and evaluate each expression. 85. x2 86. 2x2 93. 5x2 2 3y3z 101. 0.007 Use the method of Example 9 to do each calculation. Write all answers in scientific notation. 23 2 2 1m22n3p42 22 1mn22p32 4 1mn22p32 24 1mn2p2 21 87. x3 3 89. 12xz2 2 1x2z32 91. 2 z 2 y2 100. 223,470,000,000 Express each number in standard notation. 1x x 2 1x2x252 23 76. a 5x y b 3x2y23 23 22 99. 2177,000,000 22 3 24 7 4 26 Express each number in scientific notation. 97. 372,000 98. 89,500 52. 4x0 22 23 x 4 b y3 .2 Integer Exponents and Scientific Notation A25 90. 2xz z2 1x2 2 y22 92. x3z 94. 3 1x 2 z2 2 1 2 1y 2 z2 3 Use scientific notation to compute each answer. Write all answers in scientific notation. 121. Speed of sound The speed of sound in air is 3.31 3 104 centimeters per second. Compute the speed of sound in meters per minute. 122. Volume of a box Calculate the volume of a box that has dimensions of 6,000 by 9,700 by 4,700 millimeters. 123. Mass of a proton The mass of one proton is 0.00000000000000000000000167248 gram. Find the mass of one billion protons. Not For Sale 96. 125x2z232 2 5xz22 Not For Sale A Review of Basic Algebra 124. Speed of light The speed of light in a vacuum is approximately 30,000,000,000 centimeters per second. Find the speed of light in miles per hour. 1160,934.4 cm 5 1 mile.2 125. Astronomy The distance d, in miles, of the nth planet from the sun is given by the formula mass within 1 millimeter (0.04 inch) a year by using a combination of four space-based techniques. d 5 9,275,200 3 3 12n222 1 4 4 To the nearest million miles, find the distance of Earth and the distance of Mars from the sun. Give each answer in scientific notation. Jupiter Mars Earth Venus The distance from the Earth’s center to the North Pole (the polar radius) measures approximately 6,356.750 km, and the distance from the center to the equator (the equatorial radius ) measures approximately 6,378.135 km. Express each distance using scientific notation. 128. Refer to Exercise 127. Given that 1 km is approximately equal to 0.62 miles, use scientific notation to express each distance in miles. Mercury 126. License Plates The number of different license plates of the form three digits followed by three letters, as in the illustration, is 10 ? 10 ? 10 ? 26 ? 26 ? 26. Write this expression using exponents. Then evaluate it and express the result in scientific notation. WB COUNTY UTAH 12 123ABC Discovery and Writing 127. New way to the center of the earth The spectacular “blue marble’ image is the most detailed true-color image of the entire Earth to date. A new NASAdeveloped technique estimates Earth’s center of Write each expression with a single base. xm 129. xnx2 130. 3 x m 2 x x x3m15 131. 3 132. x x2 133. xm11x3 134. an23a3 4 135. Explain why 2x and 12x2 4 represent different numbers. 136. Explain why 32 3 102 is not in scientific notation. Review 137. Graph the interval 122,42 . 138. Graph the interval 12`,23 4 c 3 3,` 2 . 139. Evaluate 0 p 2 5 0 . 140. Find the distance between 27 and 25 on the number line. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Appendix NASA Goddard Space Flight Center A26 Section .3 Rational Exponents and Radicals A27 A.3 Rational Exponents and Radicals In this section, we will learn to “Dead Man’s Curve” is a 1964 hit song by the rock and roll duo Jan Berry and Dean Torrence. The song details a teenage drag race that ends in an accident. Today, dead man’s curve is a commonly used expression given to dangerous curves on our roads. Every curve has a “critical speed.” If we exceed this speed, regardless of how skilled a driver we are, we will lose control of the vehicle. The radical expression 3.9"r gives the critical speed in miles per hour when we travel a curved road with a radius of r feet. A knowledge of square roots and radicals is important and used in the construction of safe highways and roads. We will study the topic of radicals in this section. Medioimages/Photodisc/Jupiter Images © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 1.Define rational exponents whose numerators are 1. 2.Define rational exponents whose numerators are not 1. 3.Define radical expressions. 4.Simplify and combine radicals. 5.Rationalize denominators and numerators. 1. Define Rational Exponents Whose Numerators Are 1 If we apply the rule 1xm2 n 5 xmn to 1251/22 2 , we obtain 1251/22 2 5 2511/222 5 251 Caution Keep the base and multiply the exponents. 1 2 ?251 5 25 In the expression a , there is no real-number nth root of a when n is even and a , 0. For example, 12642 1/2 is not a real number, because the square of no real number is 264. 1 /n Rational Exponents Thus, 251/2 is a real number whose square is 25. Although both 52 5 25 and 1252 2 5 25, we define 251/2 to be the positive real number whose square is 25: 251/2 5 5 Read 251/2 as “the square root of 25.” In general, we have the following definition. If a $ 0 and n is a natural number, then a1/n (read as “the nth root of a”) is the nonnegative real number b such that bn 5 a Since b 5 a1/n, we have bn 5 1a1/n2 n 5 a. EXAMPLE 1Simplifying Expressions with Rational Exponents In each case, we will apply the definition of rational exponents. a. 161/2 5 4 Because 42 5 16. Read 161/2 as “the square root of 16.” b. 271/3 5 3 Because 33 5 27. Read 271/3 as “the cube root of 27.” Not For Sale c. a 1 1/4 1 b 5 3 81 1 1 1 /4 1 1 4 .” Because a b 5 . Read a b as “the fourth root of 81 3 81 81 A28 Appendix Not For Sale A Review of Basic Algebra d. 2321/5 5 2 1321/52 5 2 122 Read 321/5 as “the fifth root of 32.” Because 25 5 32. 5 22 Self Check 1 Simplify: a. 1001/2 b. 2431/5 Now Try Exercise 15. 149x22 1/2 5 7 0 x 0 Because 17 0 x 0 2 2 5 49x2 . Since x could be negative, absolute value sym- bols are necessary to guarantee that the square root is nonnegative. 116x42 1/4 5 2 0 x 0 Because 12 0 x 0 2 4 5 16x4 . Since x could be negative, absolute value sym- bols are necessary to guarantee that the fourth root is nonnegative. 1729x122 1/6 5 3x2 Because 13x22 6 5 729x12 . Since x2 is always nonnegative, no absolute value symbols are necessary. Read 1729x122 1/6 as “the sixth root of 729x12 .” If n is an odd number in the expression a1/n , the base a can be negative. EXAMPLE 2Simplifying Expressions with Rational Exponents Simplify by using the definition of rational exponent. a. 1282 1/3 5 22 b. 123,1252 1/5 5 25 c. a2 1/3 1 1 b 5 2 1,000 10 Self Check 2 Simplify: a. 121252 1/3 Now Try Exercise 19. Caution Because 1222 3 5 28. Because 1252 5 5 23,125. Because a2 1 3 1 . b 52 10 1,000 b. 12100,0002 1/5 If n is odd in the expression a1/n , we don’t need to use absolute value symbols, because odd roots can be negative. 1227x32 1/3 5 23x Because 123x2 3 5 227x3. 12128a72 1/7 5 22a Because 122a2 7 5 2128a7. We summarize the definitions concerning a1/n as follows. Summary of Definitions of a1/n If n is a natural number and a is a real number in the expression a1/n, then If a $ 0, then a1/n is the nonnegative real number b such that bn 5 a. If a , 0 e and n is odd, then a1/n is the real number b such that bn 5 a. and n is even, then a1/n is not a real number. The following chart also shows the possibilities that can occur when simplifying a1/n . © Cengage Learning. All rights reserved. No distribution allowed without express authorization. If n is even in the expression a1/n and the base contains variables, we often use absolute value symbols to guarantee that an even root is nonnegative. Section © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Strategy for Simplifying Expressions of the Form a1/n .3 Rational Exponents and Radicals A29 n a1/n a50 n is a natural number. 01/n is the real number 0 because 0n 5 0. 01/2 5 0 because 02 5 0. 01/5 5 0 because 05 5 0. a.0 n is a natural number. 161/2 5 4 because 42 5 16. 271/3 5 3 because 33 5 27. a,0 n is an odd natural number. a1/n is the nonnegative real number such that 1a1/n2 n 5 a. a,0 n is an even natural number. a1/n is not a real number. 1292 1/2 is not a real number. 12812 1/4 is not a real number. a Examples a1/n is the real number such that 1a1/n2 n 5 a. 12322 1/5 5 22 because 1222 5 5 232. 121252 1/3 5 25 because 1252 3 5 2125. 2. Define Rational Exponents Whose Rational Exponents Are Not 1 The definition of a1/n can be extended to include rational exponents whose numerators are not 1. For example, 43/2 can be written as either 141/22 3 or 1432 1/2 Because of the Power Rule, 1xm2 n 5 xmn. This suggests the following rule. Rule for Rational Exponents If m and n are positive integers, the fraction mn is in lowest terms, and a1/n is a real number, then am/n 5 1a1/n2 m 5 1am2 1/n In the previous rule, we can view the expression am/n in two ways: 1. 1a1/n2 m: the mth power of the nth root of a 2. 1am2 1/n: the nth root of the mth power of a For example, 12272 2/3 can be simplified in two ways: 12272 2/3 5 3 12272 1/3 4 2 5 1232 2 59 12272 2/3 5 3 12272 2 4 1/3 59 or 5 17292 1/3 Not For Sale As this example suggests, it is usually easier to take the root of the base first to avoid large numbers. Appendix A Not For Sale A Review of Basic Algebra Comment Negative Rational Exponents It is helpful to think of the phrase power over root when we see a rational exponent. The numerator of the fraction represents the power and the denominator represents the root. Begin with the root when simplifying to avoid large numbers. If m and n are positive integers, the fraction mn is in lowest terms and a1/n is a real number, then 1 1 1a 2 02 a2m/n 5 m/n and 2m/n 5 am/n a a EXAMPLE 3 Simplifying Expressions with Rational Exponents We will apply the rules for rational exponents. a. 253/2 5 1251/22 3 b. a2 5 53 5 125 x6 2/3 x6 1/3 2 b 5 c a2 b d 1,000 1,000 5 a2 5 x2 2 b 10 x4 100 1 1 d. 23/4 5 813/4 2/5 32 81 1 5 5 1811/42 3 1321/52 2 c. 3222/5 5 1 22 1 5 4 5 33 5 Self Check 3 Simplify: a. 493/2 b. 1623/4 Now Try Exercise 43. c. 5 27 1 127x32 22/3 Because of the definition, rational exponents follow the same rules as integer exponents. EXAMPLE 4 Using Exponent Rules to Simplify Expressions with Rational Exponents Simplify each expression. Assume that all variables represent positive numbers, and write answers without using negative exponents. a. 136x2 1/2 5 361/2x1/2 b. 5 6x1/2 1a1/3b2/32 6 a6/3b12/3 5 3 2 1y 2 y6 5 a2b4 y6 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A30 Section A.3 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. ax/2ax/4 c. x/6 5 ax/21x/42x/6 a Rational Exponents and Radicals d. c A31 2c22/5 5/3 d 5 12c22/524/52 5/3 c4/5 5 a6x/1213x/1222x/12 5 a7x/12 5 3 1212 1c26/52 4 5/3 5 21c230/15 5 2c22 1 5 2 2 c 5 1212 5/3 1c26/52 5/3 Self Check 4 Use the directions for Example 4: 19r2s2 1/2 y2 1/2 b3/7b2/7 a. a b b. c. 49 b4/7 rs23/2 Now Try Exercise 59. 3. Define Radical Expressions Radical signs can also be used to express roots of numbers. n Definition of "a If n is a natural number greater than 1 and if a1/n is a real number, then n "a 5 a1/n n In the radical expression "a, the symbol " is the radical sign, a is the radicand, and n is the index (or the order) of the radical expression. If the order is 2, the expression is a square root, and we do not write the index. 2 "a 5 " a If the index of a radical is 3, we call the radical a cube root. nth Root of a Nonnegative Number n If n is a natural number greater than 1 and a $ 0, then "a is the nonnegative number whose nth power is a. n n Q"aR 5 a Caution n In the expression "a, there is no real-number nth root of a when n is even and a , 0. For example, "264 is not a real number, because the square of no real number is 264. n n If 2 is substituted for n in the equation Q"aR 5 a, we have 2 2 2 Q" aR 5 Q"aR 5 "a"a 5 a for a $ 0 This shows that if a number a can be factored into two equal factors, either of those factors is a square root of a. Furthermore, if a can be factored into n equal factors, any one of those factors is an nth root of a. Not For Sale n If n is an odd number greater than 1 in the expression "a, the radicand can be negative. A32 Appendix Not For Sale A Review of Basic Algebra EXAMPLE 5 Finding nth Roots of Real Numbers We apply the definitions of cube root and fifth root. 3 b. " 28 5 22 c. 27 3 5 2 Å 1,000 10 3 2 Because 1232 3 5 227. Because 1222 3 5 28. Because a2 5 5 2243 5 2Q" 2243R d. 2" 1 5 2 232 53 3 Self Check 5 Find each root: a. " 216 b. Now Try Exercise 69. 27 3 3 . b 52 10 1,000 1 Å 32 5 2 n We summarize the definitions concerning "a as follows. n Summary of Definitions of "a If n is a natural number greater than 1 and a is a real number, then n n n If a $ 0, then "a is the nonnegative real number such that Q"aR 5 a. n If a , 0 e n n and n is odd, then "a is the real number such that Q"aR 5 a. n and n is even, then "a is not a real number. The following chart also shows the possibilities that can occur when simplifyn ing "a. Strategy for Simplifying n Expressions of the Form "a a n n Examples "a a50 n is a natural "0 is the real number number 0 greater than because 0n 5 0. 1. "0 5 0 because 03 5 0. 5 "0 5 0 because 05 5 0. a.0 n is a natural number greater than 1. "16 5 4 because 42 5 16. 3 " 27 5 3 because 33 5 27. a,0 a,0 n n "a is the nonnegative real number such n n that Q"aR 5 a. 3 n 5 n is an odd "a is the real " 232 5 22 because 1222 5 5 232. 3 natural num- number such "2125 5 25 because 1252 3 5 2125. n n ber greater that Q"aR 5 a. than 1. n is an even natural number. n "a is not a real "29 is not a real number. 4 number. " 281 is not a real number. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 3 a. "227 5 23 Section .3 Rational Exponents and Radicals A33 We have seen that if a1/n is real, then am/n 5 1a1/n2 m 5 1am2 1/n . This same fact can be stated in radical notation. n n am/n 5 1"a2 m 5 "am Thus, the mth power of the nth root of a is the same as the nth root of the mth power 3 of a. For example, to find " 272, we can proceed in either of two ways: 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 3 3 3 3 272 5 " 729 5 9 " 272 5 Q" 27R 5 32 5 9 or " By definition, "a2 represents a nonnegative number. If a could be negative, we must use absolute value symbols to guarantee that "a2 will be nonnegative. Thus, if a is unrestricted, "a2 5 0 a 0 A similar argument holds when the index is any even natural number. The sym4 bol "a4, for example, means the positive fourth root of a4. Thus, if a is unrestricted, 4 4 " a 5 0a0 EXAMPLE 6Simplifying Radical Expressions 6 3 3 If x is unrestricted, simplify a. " 64x6 b. " x c. "9x8 SOLUTION We apply the definitions of sixth roots, cube roots, and square roots. 6 a. "64x6 5 2 0 x 0 3 3 b. " x 5 x Use absolute value symbols to guarantee that the result will be nonnegative. Because the index is odd, no absolute value symbols are needed. c. "9x8 5 3x4 Because 3x4 is always nonnegative, no absolute value symbols are needed. Self Check 6 Use the directions for Example 6: 4 a. " 16x4 3 b. " 27y3 Now Try Exercise 73. 4 8 c. " x 4. Simplify and Combine Radicals Many properties of exponents have counterparts in radical notation. For example, 1 /n since a1/nb1/n 5 1ab2 1/n and ab1/n 5 1ab2 1/n and 1b 2 02 , we have the following. Multiplication and Division Properties of Radicals If all expressions represent real numbers, n n n "a"b 5 "ab n "a n "b 5 a 1b 2 02 Åb n In words, we say Not For Sale The product of two nth roots is equal to the nth root of their product. The quotient of two nth roots is equal to the nth root of their quotient. Appendix Not For Sale A Review of Basic Algebra Caution These properties involve the nth root of the product of two numbers or the nth root of the quotient of two numbers. There is no such property for sums or differences. For example, "9 1 4 2 "9 1 "4, because "9 1 4 5 "13 but "9 1 "4 5 3 1 2 5 5 and "13 2 5. In general, "a 1 b 5 "a 1 "b and "a 2 b 5 "a 2 "b Numbers that are squares of positive integers, such as 1, 4, 9, 16, 25, and 36 are called perfect squares. Expressions such as 4x2 and 19x6 are also perfect squares, because each one is the square of another expression with integer exponents and rational coefficients. 2 1 1 4x2 5 12x2 2 and x6 5 a x3 b 9 3 Numbers that are cubes of positive integers, such as 1, 8, 27, 64, 125, and 216 1 9 x are also perfect cubes, are called perfect cubes. Expressions such as 64x3 and 27 because each one is the cube of another expression with integer exponents and rational coefficients. 64x3 5 14x2 3 and 3 1 9 1 x 5 a x3 b 3 27 There are also perfect fourth powers, perfect fifth powers, and so on. We can use perfect powers and the Multiplication Property of Radicals to simplify many radical expressions. For example, to simplify "12x5, we factor 12x5 so that one factor is the largest perfect square that divides 12x5. In this case, it is 4x4. We then rewrite 12x5 as 4x4 ? 3x and simplify. "12x5 5 "4x4 ? 3x Factor 12x5 as 4x4 ? 3x. 5 "4x4"3x Use the Multiplication Property of Radicals: "ab 5 "a"b. 5 2x2"3x "4x4 5 2x2 3 To simplify " 432x9y, we find the largest perfect-cube factor of 432x9y (which is 216x9) and proceed as follows: 3 3 "432x9y 5 "216x9 ? 2y Factor 432x9y as 216x9 ? 2y. 3 3 3 3 3 5" 216x9 " 2y Use the Multiplication Property of Radicals: " ab 5 " a" b. 3 5 6x3" 2y 3 " 216x9 5 6x3 Radical expressions with the same index and the same radicand are called like or similar radicals. We can combine the like radicals in 3"2 1 2"2 by using the Distributive Property. 3"2 1 2"2 5 13 1 22 "2 5 5"2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A34 Section .3 Rational Exponents and Radicals A35 This example suggests that to combine like radicals, we add their numerical coefficients and keep the same radical. When radicals have the same index but different radicands, we can often change them to equivalent forms having the same radicand. We can then combine them. For example, to simplify "27 2 "12, we simplify both radicals and combine like radicals. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. "27 2 "12 5 "9 ? 3 2 "4 ? 3 Factor 27 and 12. 5 "9"3 2 "4"3 5 3"3 2 2"3 "ab 5 "a"b 5 "3 Combine like radicals. "9 5 3 and "4 5 2. EXAMPLE 7Adding and Subtracting Radical Expressions 5 5 Simplify: a. "50 1 "200 b. 3z" 64z 2 2" 2z6 SOLUTION We will simplify each radical expression and then combine like radicals. a. "50 1 "200 5 "25 ? 2 1 "100 ? 2 5 "25"2 1 "100"2 5 15"2 5 5"2 1 10"2 5 5 5 5 5 b. 3z" 64z 2 2" 2z6 5 3z" 32 ? 2z 2 2" z ? 2z 5 5 5 5 5 5 3z" 32 " 2z 2 2" z "2z 5 5 5 3z 122 " 2z 2 2z" 2z 5 5 2z 5 6z" 2z 2 2z" 5 5 4z" 2z Self Check 7 Simplify: a. "18 2 "8 3 3 b. 2" 81a4 1 a" 24a Now Try Exercise 85. 5. Rationalize Denominators and Numerators By rationalizing the denominator, we can write a fraction such as "5 "3 as a fraction with a rational number in the denominator. All that we must do is multiply both the numerator and the denominator by "3. (Note that "3"3 is the rational number 3.) "5 "3 5 "5"3 "3"3 5 "15 3 To rationalize the numerator, we multiply both the numerator and the denominator by "5. (Note that "5"5 is the rational number 5.) Not For Sale "5 "3 5 "5"5 "3"5 5 5 "15 A36 Appendix Not For Sale A Review of Basic Algebra EXAMPLE 8Rationalizing the Denominator of a Radical Expression Rationalize each denominator and simplify. Assume that all variables represent positive numbers. a. 1 "7 b. 3 3 3a3 c. d. Å4 Åx Å 5x5 3 SOLUTION We will multiply both the numerator and the denominator by a radical that will make the denominator a rational number. 1 "7 5 5 1"7 "7"7 "7 7 b. 3 " 3 3 5 3 Å4 "4 3 5 5 5 c. 3 "3 5 Åx "x d. 3 3 " 3" 2 3 3 "4"2 Multiply numerator and 3 denominator by "2, because 3 6 " 3 3 " 8 3 " 6 2 3a3 "3a3 5 5 Å 5x "5x5 "3a3"5x "3x x 5 5 "a2"15ax 5x3 5 a"15ax 5x3 5 5 "x"x 3 "4"2 5 "8 5 2. "3"x 3 5 "5x5"5x "15a3x Multiply numerator and denominator by "5x, because "5x5 "5x 5 "25x6 5 5x3. "25x6 Self Check 8 Use the directions for Example 8: a. 6 "6 b. 3 2 Å 5x Now Try Exercise 101. EXAMPLE 9Rationalizing the Numerator of a Radical Expression Rationalize each numerator and simplify. Assume that all variables represent posi3 "x 2" 9x tive numbers: a. b. 7 3 SOLUTION We will multiply both the numerator and the denominator by a radical that will make the numerator a rational number. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. a. Section a. "x "x ? "x 5 7 7"x x b. .3 Rational Exponents and Radicals A37 3 3 3 2" 9x 2" 9x ? " 3x2 5 3 3 3" 3x2 5 5 5 5 7"x © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Self Check 9 Use the directions for Example 9: a. "2x 5 b. 3 2" 27x3 3 3" 3x2 2 13x2 3 3" 3x2 2x 3 "3x2 Divide out the 3’s. 3 3" 2y2 6 Now Try Exercise 111. After rationalizing denominators, we often can simplify an expression. EXAMPLE 10Rationalizing Denominators and Simplifying Simplify: 1 1 1 . Å2 Å8 SOLUTION We will rationalize the denominators of each radical and then combine like radicals. 1 1 1 1 1 5 1 Å2 Å8 "2 "8 5 5 5 5 Self Check 10 Simplify: 1"2 "2"2 1 1"2 1 "1 1 1 "1 1 ; 5 5 5 5 Å2 Å 8 "2 "2 "8 "8 "8"2 "2 "2 1 2 "16 "2 "2 1 2 4 3"2 4 x 3 x 2 3 . Å2 Å 16 Now Try Exercise 115. Another property of radicals can be derived from the properties of exponents. If all of the expressions represent real numbers, n m n mn #"x 5 "x1/m 5 1x1/m2 1/n 5 x1/ 1mn2 5 "x m m mn n #"x 5 "x1/n 5 1x1/n2 1/m 5 x1/1nm2 5 "x Not For Sale These results are summarized in the following theorem (a fact that can be proved). A38 Appendix Not For Sale A Review of Basic Algebra Theorem If all of the expressions involved represent real numbers, then m n mn n m #"x 5 #"x 5 "x We can use the previous theorem to simplify many radicals. For example, 3 3 #"8 5 #"8 5 "2 EXAMPLE 11Simplifying Radicals Using the Previously Stated Theorem Simplify. Assume that x and y are positive numbers. 12 9 6 a. "4 b. "x3 c. "8y3 SOLUTION In each case, we will write the radical as an exponential expression, simplify the resulting expression, and write the final result as a radical. 6 3 a. "4 5 41/6 5 1222 1/6 5 22/6 5 21/3 5 "2 12 4 b. "x3 5 x3/12 5 x1/4 5 " x 9 3 c. " 8y3 5 123y32 1/9 5 12y2 3/9 5 12y2 1/3 5 " 2y 4 Self Check 11 Simplify: a. " 4 9 b. " 27x3 Now Try Exercise 117. 1 Self Check Answers 1. a. 10 b. 3 2. a. 25 b. 210 3. a. 343 b. 8 y 1 c. 9x2 4. a. b. b1/7 c. 3s2 5. a. 6 b. 2 6. a. 2 0 x 0 7 2 3 " 50x2 3 b. 3y c. x2 7. a. "2 b. 8a"3a 8. a. "6 b. 5x 3 y " 4x 2x 3 b. 3 10. 11. a. "2 b. "3x 9. a. 4 "4y 5"2x Exercises A.3 Getting Ready 3. If a , 0 and n is an even number, then a1/n is a real number. You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. If a 5 0 and n is a natural number, then a1/n 5 2. If a . 0 and n is a natural number, then a1/n is a number. 4. 62/3 can be written as n . 6. "a 5 n 8. 7. "a"b 5 9. "x 1 y m n . 2 5. "a 5 n or n "x 1 "y m a 5 Åb n 10. #"x or #"x can be written as . © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Rational exponents can be used to simplify many radical expressions, as shown in the following example. Section Practice Simplify each expression. 11. 91/2 1 1/2 13. a b 25 12. 81/3 16 1/4 14. a b 625 15. 2811/4 16. 2a © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 17. 110,0002 1/4 19. a2 8 1/3 b 27 18. 1,0241/5 27 1/3 b 8 20. 2641/3 21. 12642 1/2 22. 121252 1/3 Simplify each expression. Use absolute value symbols when necessary. 23. 116a22 1/2 24. 125a42 1/2 25. 116a42 1/4 27. 1232a52 1/5 29. 12216b 2 6 1/3 31. a 16a b 25b2 33. a2 30. 1256t 2 8 1/4 1/2 4 26. 1264a32 1/3 28. 164a62 1/6 1,000x6 1/3 b 27y3 32. a2 34. a a5 1/5 b 32b10 49t2 1/2 b 100z4 Simplify each expression. Write all answers without using negative exponents. 35. 43/2 3/2 37. 216 39. 21,0002/3 41. 6421/2 40. 1003/2 42. 2521/2 43. 6423/2 44. 4923/2 45. 2923/2 4 5/2 47. a b 9 49. a2 36. 82/3 38. 1282 2/3 46. 12272 22/3 48. a 27 22/3 b 64 50. a 25 3/2 b 81 125 24/3 b 8 Simplify each expression. Assume that all variables represent positive numbers. Write all answers without using negative exponents. 51. 1100s42 1/2 52. 164u6v32 1/3 53. 132y10z52 21/5 55. 1x10y52 3/5 57. 1r8s162 23/4 59. a2 54. 1625a4b82 21/4 56. 164a6b122 5/6 58. 128x9y122 22/3 61. a 63. .3 Rational Exponents and Radicals A39 27r6 22/3 b 1,000s12 a2/5a4/5 a1/5 62. a2 64. 60. a 16x4 3/4 b 625y8 x6/7x3/7 x2/7x5/7 Simplify each radical expression. 65. "49 66. "81 3 3 67. " 125 68. " 264 3 5 69. " 2125 70. " 2243 32 256 71. 5 2 72. 4 Å 100,000 Å 625 Simplify each expression, using absolute value symbols when necessary. Write answers without using negative exponents. 73. "36x2 75. "9y4 3 77. " 8y3 79. x4y8 Å z12 4 74. 2"25y2 76. "a4b8 3 78. " 227z9 80. a10b5 Å c15 5 Simplify each expression. Assume that all variables represent positive numbers so that no absolute value symbols are needed. 81. "8 2 "2 82. "75 2 2"27 2 2 83. "200x 1 "98x 84. "128a3 2 a"162a 85. 2"48y5 2 3y"12y3 86. y"112y 1 4"175y3 3 3 87. 2" 81 1 3" 24 4 4 5 89. "768z 1 "48z5 4 4 88. 3" 32 2 2" 162 5 5 2 90. 22"64y 1 3" 486y2 91. "8x2y 2 x"2y 1 "50x2y 92. 3x"18x 1 2"2x3 2 "72x3 3 3 3 93. " 16xy4 1 y" 2xy 2 " 54xy4 4 4 4 94. " 512x5 2 " 32x5 1 " 1,250x5 Rationalize each denominator and simplify. Assume that all variables represent positive numbers. 95. 97. 3 "3 2 96. 98. "x 2 99. 3 100. "2 5a 101. 3 102. "25a 103. 2b Not For Sale 8a6 2/3 b 125b9 32m10 22/5 b 243n15 105. 4 2 "3a 2u4 Å 9v 3 104. 106. 6 "5 8 "y 4d 3 " 9 7 3 " 36c x Å 2y 3s5 Å 4r2 3 2 Not For Sale A Review of Basic Algebra Rationalize each numerator and simplify. Assume that all variables are positive numbers. "5 "y 107. 108. 10 3 3 109. 111. "9 3 3 110. 5 " 16b3 64a 112. 2 "16b 16 3x Å 57 Rationalize each denominator and simplify. 113. 115. 1 1 2 Å 3 Å 27 114. x x x 2 1 Å2 Å 32 Å8 116. 4 1 1 1 3 Å 2 Å 16 3 y y y 1 3 2 3 Å 4 Å 32 Å 500 3 6 118. " 27 10 119. "16x6 6 120. " 27x9 Discovery and Writing We often can multiply and divide radicals with different indi3 ces. For example, to multiply "3 by " 5, we first write each radical as a sixth root 6 6 3 3 5" 27 "3 5 31/2 5 33/6 5 " 3 6 2 6 " 5 5 51/3 5 52/6 5 " 5 5" 25 and then multiply the sixth roots. 3 6 6 6 3 121. "2" 2 123. 4 " 3 6 3 122. "3" 5 124. 3 " 2 "5 "2 4 4 125. For what values of x does " x 5 x? Explain. 126. If all of the radicals involved represent real numbers and y 2 0, explain why n x "x 5 n Åy "y 127. If all of the radicals involved represent real numbers and there is no division by 0, explain why n Simplify each radical expression. 117. "9 Use this idea to write each of the following expressions as a single radical. m x 2m/n n y 5 a b y Å xm n 128. The definition of xm/n requires that "x be a real number. Explain why this is important. (Hint: Consider what happens when n is even, m is odd, and x is negative.) Review 1 29. Write 22 , x # 5 using interval notation. 130. Write the expression 0 3 2 x 0 without using absolute value symbols. Assume that x . 4. Evaluate each expression when x 5 22 and y 5 3. xy 1 4y 131. x2 2 y2 132. x 133. Write 617,000,000 in scientific notation. 134. Write 0.00235 3 104 in standard notation. "3"5 5 "27"25 5 " 1272 1252 5 "675 Division is similar. A.4 Polynomials In this section, we will learn to 1.Define polynomials. 2.Add and subtract polynomials. 3.Multiply polynomials. 4.Rationalize denominators. 5.Divide polynomials. © PCN Photography/Alamy Football is one of the most popular sports in the United States. Brett Favre, one of the most talented quarterbacks ever to play the game, holds the record for the most career NFL touchdown passes, the most NFL pass completions, and the most passing yards. He led the Green Bay Packers to the Super Bowl and most recently played for the Minnesota Vikings. His nickname is Gunslinger. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Appendix A40 Section .4 Polynomials A41 An algebraic expression can be used to model the trajectory or path of a football when passed by Brett Favre. Suppose the height in feet, t seconds after the football leaves Brett’s hand, is given by the algebraic expression 20.1t2 1 t 1 5.5 At a time of t 5 3 seconds, we see that the height of the football is 20.1 132 2 1 3 1 5.5 5 7.6 ft. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Algebraic expressions like 20.1t2 1 t 1 5.5 are called polynomials, and we will study them in this section. 1. Define Polynomials A monomial is a real number or the product of a real number and one or more variables with whole-number exponents. The number is called the coefficient of the variables. Some examples of monomials are 3x, 7ab2, 25ab2c4, x3, and 212 with coefficients of 3, 7, 25, 1, and 212, respectively. The degree of a monomial is the sum of the exponents of its variables. All nonzero constants (except 0) have a degree of 0. The degree of 3x is 1. The degree of 7ab2 is 3. The degree of 25ab2c4 is 7. The degree of x3 is 3. The degree of 212 is 0 (since 212 5 212x0). 0 has no defined degree. A monomial or a sum of monomials is called a polynomial. Each monomial in that sum is called a term of the polynomial. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial. Monomials 2 Binomials Trinomials 3x 2a 1 3b x2 1 7x 2 4 225xy 4x3 2 3x2 4y4 2 2y 1 12 a2b3c 22x3 2 4y2 12x3y2 2 8xy 2 24 The degree of a polynomial is the degree of the term in the polynomial with highest degree. The only polynomial with no defined degree is 0, which is called the zero polynomial. Here are some examples. • 3x2y3 1 5xy2 1 7 is a trinomial of 5th degree, because its term with highest degree (the first term) is 5. • 3ab 1 5a2b is a binomial of degree 3. 4 3z4 2 "7 is a polynomial, because its variables have whole• 5x 1 3y2 1 " number exponents. It is of degree 4. 5 • 27y1/2 1 3y2 1 " 3z is not a polynomial, because one of its variables (y in the first term) does not have a whole-number exponent. If two terms of a polynomial have the same variables with the same exponents, they are like or similar terms. To combine the like terms in the sum 3x2y 1 5x2y or the difference 7xy2 2 2xy2, we use the Distributive Property: 3x2y 1 5x2y 5 13 1 52 x2y 5 8x2y 7xy2 2 2xy2 5 17 2 22 xy2 Not For Sale 5 5xy2 This illustrates that to combine like terms, we add (or subtract) their coefficients and keep the same variables and the same exponents. A42 Appendix Not For Sale A Review of Basic Algebra 2. Add and Subtract Polynomials Recall that we can use the Distributive Property to remove parentheses enclosing the terms of a polynomial. When the sign preceding the parentheses is +, we simply drop the parentheses: 1 1a 1 b 2 c2 5 11 1a 1 b 2 c2 5 1a 1 1b 2 1c When the sign preceding the parentheses is 2, we drop the parentheses and the 2 sign and change the sign of each term within the parentheses. 2 1a 1 b 2 c2 5 21 1a 1 b 2 c2 5 21a 1 1212 b 2 1212 c 5 2a 2 b 1 c We can use these facts to add and subtract polynomials. To add (or subtract) polynomials, we remove parentheses (if necessary) and combine like terms. EXAMPLE 1Adding Polynomials Add: 13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2 . Solution To add the polynomials, we remove parentheses and combine like terms. 13x3y 1 5x2 2 2y2 1 12x3y 2 5x2 1 3x2 5 3x3y 1 5x2 2 2y 1 2x3y 2 5x2 1 3x 5 3x3y 1 2x3y 1 5x2 2 5x2 2 2y 1 3x Use the Commutative Property to rearrange terms. 5 5x3y 2 2y 1 3x Combine like terms. We can add the polynomials in a vertical format by writing like terms in a column and adding the like terms, column by column. 3x3y 1 5x2 2 2y 2x3y 2 5x2 1 3x 3 5x y 2 2y 1 3x Self Check 1 Add: 14x2 1 3x 2 52 1 13x2 2 5x 1 72 . Now Try Exercise 21. EXAMPLE 2Subtracting Polynomials Subtract: 12x2 1 3y22 2 1x2 2 2y2 1 72 . Solution To subtract the polynomials, we remove parentheses and combine like terms. 12x2 1 3y22 2 1x2 2 2y2 1 72 5 2x2 1 3y2 2 x2 1 2y2 2 7 5 2x2 2 x2 1 3y2 1 2y2 2 7 Use the Commutative Property to rearrange terms. 5 x2 1 5y2 2 7 Combine like terms. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 5a1b2c Section .4 Polynomials A43 We can subtract the polynomials in a vertical format by writing like terms in a column and subtracting the like terms, column by column. 2x2 1 3y2 2 1x2 2 2y2 1 72 x2 1 5y2 2 7 2x2 2 x2 5 12 2 12 x2 3y2 2 1222 y2 5 3y2 1 2y2 5 13 1 22 y2 0 2 7 5 27 Self Check 2 Subtract: 14x2 1 3x 2 52 2 13x2 2 5x 1 72 . © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Now Try Exercise 23. We can also use the Distributive Property to remove parentheses enclosing several terms that are multiplied by a constant. For example, 4 13x2 2 2x 1 62 5 4 13x22 2 4 12x2 1 4 162 5 12x2 2 8x 1 24 This example suggests that to add multiples of one polynomial to another, or to subtract multiples of one polynomial from another, we remove parentheses and combine like terms. EXAMPLE 3 Using the Distributive Property and Combining Like Terms Simplify: 7x 12y2 1 13x22 2 5 1xy2 2 13x32 . Solution 7x 12y2 1 13x22 2 5 1xy2 2 13x32 5 14xy2 1 91x3 2 5xy2 1 65x3 Use the Distributive Property to remove parentheses. 5 14xy 2 5xy 1 91x 1 65x Use the Commutative Property to rearrange terms. 5 9xy2 1 156x3 Combine like terms. 2 2 3 3 Self Check 3 Simplify: 3 12b2 2 3a2b2 1 2b 1b 1 a22 . Now Try Exercise 30. 3. Multiply Polynomials To find the product of 3x2y3z and 5xyz2, we proceed as follows: 13x2y3z2 15xyz22 5 3 ? x2 ? y3 ? z ? 5 ? x ? y ? z2 5 3 ? 5 ? x2 ? x ? y3 ? y ? z ? z2 Use the Commutative Property to rearrange terms. 5 15x3y4z3 This illustrates that to multiply two monomials, we multiply the coefficients and then multiply the variables. To find the product of a monomial and a polynomial, we use the Distributive Property. 3xy2 12xy 1 x2 2 7yz2 5 3xy2 12xy2 1 13xy22 1x22 2 13xy22 17yz2 5 6x2y3 1 3x3y2 2 21xy3z Not For Sale This illustrates that to multiply a polynomial by a monomial, we multiply each term of the polynomial by the monomial. A44 Appendix Not For Sale A Review of Basic Algebra To multiply one binomial by another, we use the Distributive Property twice. EXAMPLE 4 Multiplying Binomials Multiply: a. 1x 1 y2 1x 1 y2 b. 1x 2 y2 1x 2 y2 c. 1x 1 y2 1x 2 y2 5 x2 1 xy 1 xy 1 y2 5 x2 1 2xy 1 y2 b. 1x 2 y2 1x 2 y2 5 1x 2 y2 x 2 1x 2 y2 y 5 x2 2 xy 2 xy 1 y2 5 x2 2 2xy 1 y2 c. 1x 1 y2 1x 2 y2 5 1x 1 y2 x 2 1x 1 y2 y 5 x2 1 xy 2 xy 2 y2 5 x2 2 y2 Self Check 4 Multiply: a. 1x 1 22 1x 1 22 c. 1x 1 42 1x 2 42 b. 1x 2 32 1x 2 32 Now Try Exercise 45. The products in Example 4 are called special products. Because they occur so often, it is worthwhile to learn their forms. Special Product Formulas 1x 1 y2 2 5 1x 1 y2 1x 1 y2 5 x2 1 2xy 1 y2 1x 2 y2 2 5 1x 2 y2 1x 2 y2 5 x2 2 2xy 1 y2 1x 1 y2 1x 2 y2 5 x2 2 y2 Caution Remember that 1x 1 y2 2 and 1x 2 y2 2 have trinomials for their products and that For example, 1x 1 y2 2 2 x2 1 y2 and 1x 2 y2 2 2 x2 2 y2 13 1 52 2 2 32 1 52 and 13 2 52 2 2 32 2 52 82 2 9 1 25 64 2 34 1222 2 2 9 2 25 4 2 216 We can use the FOIL method to multiply one binomial by another. FOIL is an acronym for First terms, Outer terms, Inner terms, and Last terms. To use this method to multiply 3x 2 4 by 2x 1 5, we write © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Solution a. 1x 1 y2 1x 1 y2 5 1x 1 y2 x 1 1x 1 y2 y Section .4 Polynomials A45 First terms Last terms 13x 2 42 12x 1 52 5 3x 12x2 1 3x 152 2 4 12x2 2 4 152 Inner terms 5 6x2 1 15x 2 8x 2 20 Outer terms 5 6x2 1 7x 2 20 In this example, © Cengage Learning. All rights reserved. No distribution allowed without express authorization. • • • • the product of the product of the product of the product of the first terms is 6x2, the outer terms is 15x, the inner terms is 28x, and the last terms is 220. The resulting like terms of the product are then combined. EXAMPLE 5 Using the FOIL Method to Multiply Polynomials Use the FOIL method to multiply: Q"3 1 xRQ2 2 "3xR. Solution 1"3 1 x2 12 2 "3x2 5 2"3 2 "3"3x 1 2x 2 x"3x 5 2"3 2 3x 1 2x 2 "3x2 5 2"3 2 x 2 "3x2 Self Check 5 Multiply: Q2x 1 "3RQx 2 "3R. Now Try Exercise 63. To multiply a polynomial with more than two terms by another polynomial, we multiply each term of one polynomial by each term of the other polynomial and combine like terms whenever possible. EXAMPLE 6 Multiplying Polynomials Multiply: a. 1x 1 y2 1x2 2 xy 1 y22 b. 1x 1 32 3 Solution a. 1x 1 y2 1x2 2 xy 1 y22 5 x3 2 x2y 1 xy2 1 yx2 2 xy2 1 y3 5 x3 1 y3 b. 1x 1 32 3 5 1x 1 32 1x 1 32 2 5 1x 1 32 1x2 1 6x 1 92 5 x3 1 6x2 1 9x 1 3x2 1 18x 1 27 5 x3 1 9x2 1 27x 1 27 Self Check 6 Multiply: 1x 1 22 12x2 1 3x 2 12 . Now Try Exercise 67. We can use a vertical format to multiply two polynomials, such as the polynomials given in Self Check 6. We first write the polynomials as follows and draw a line beneath them. We then multiply each term of the upper polynomial by each term of the lower polynomial and write the results so that like terms appear in each column. Finally, we combine like terms column by column. Not For Sale Appendix Not For Sale A Review of Basic Algebra 2x2 1 3x 2 1 x12 4x2 1 6x 2 2 2x 1 3x2 2 x Multiply 2x2 1 3x 2 1 by 2. 2x3 1 7x2 1 5x 2 2 In each column, combine like terms. 3 Multiply 2x2 1 3x 2 1 by x. If n is a whole number, the expressions an 1 1 and 2an 2 3 are polynomials and we can multiply them as follows: 1an 1 12 12an 2 32 5 2a2n 2 3an 1 2an 2 3 5 2a2n 2 an 2 3 Combine like terms. We can also use the methods previously discussed to multiply expressions that are not polynomials, such as x22 1 y and x2 2 y21. 1x22 1 y2 1x2 2 y212 5 x2212 2 x22y21 1 x2y 2 y121 1 1 x2y 2 y0 x2y 1 5 1 2 2 1 x2y 2 1 xy 5 x0 2 5 x2y 2 x0 5 1 and y0 5 1. 1 x2y 4. Rationalize Denominators If the denominator of a fraction is a binomial containing square roots, we can use the product formula 1x 1 y2 1x 2 y2 to rationalize the denominator. For example, to rationalize the denominator of 6 "7 1 2 To rationalize a denominator means to change the denominator into a rational number. we multiply the numerator and denominator by "7 2 2 and simplify. 6 "7 1 2 5 6 1"7 2 22 1"7 1 22 1"7 2 22 6 1"7 2 22 724 6 1"7 2 22 5 3 5 "7 2 2 "7 2 2 51 Here the denominator is a rational number. 5 2 1"7 2 22 In this example, we multiplied both the numerator and the denominator of the given fraction by "7 2 2. This binomial is the same as the denominator of the given fraction "7 1 2, except for the sign between the terms. Such binomials are called conjugate binomials or radical conjugates. Conjugate Binomials Conjugate binomials are binomials that are the same except for the sign between their terms. The conjugate of a 1 b is a 2 b, and the conjugate of a 2 b is a 1 b. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A46 Section .4 Polynomials A47 EXAMPLE 7Rationalizing the Denominator of a Radical Expression Rationalize the denominator: "3x 2 "2 "3x 1 "2 1x . 02 . Solution We multiply the numerator and the denominator by "3x 2 "2 (the conjugate of "3x 1 "22 and simplify. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. "3x 2 "2 "3x 1 "2 5 Q"3x 2 "2RQ"3x 2 "2R 5 "3x"3x 2 "3x"2 2 "2"3x 1 "2"2 5 3x 2 "6x 2 "6x 1 2 3x 2 2 5 3x 2 2"6x 1 2 3x 2 2 Q"3x 1 "2RQ"3x 2 "2R 2 "3x 2 "2 "3x 2 "2 51 2 Q"3xR 2 Q"2R Self Check 7 Rationalize the denominator: Now Try Exercise 89. "x 1 2 "x 2 2 . In calculus, we often rationalize a numerator. EXAMPLE 8Rationalizing the Numerator of a Radical Expression Rationalize the numerator: "x 1 h 2 "x . h Solution To rid the numerator of radicals, we multiply the numerator and the denominator by the conjugate of the numerator and simplify. Q"x 1 h 2 "xRQ"x 1 h 1 "xR "x 1 h 2 "x 5 h hQ"x 1 h 1 "xR 5 5 5 x1h2x "x 1 h 1 "x Here the numerator has no radicals. Divide out the common factor of h. hQ"x 1 h 1 "xR h hQ"x 1 h 1 "xR 1 "x 1 h 1 "x Self Check 8 Rationalize the numerator: "4 1 h 2 2 . h Not For Sale Now Try Exercise 99. "x 1 h 1 "x 51 Appendix Not For Sale A Review of Basic Algebra 5. Divide Polynomials To divide monomials, we write the quotient as a fraction and simplify by using the rules of exponents. For example, 6x2y3 5 23x223y321 22x3y 5 23x21y2 3y2 x To divide a polynomial by a monomial, we write the quotient as a fraction, write the fraction as a sum of separate fractions, and simplify each one. For example, to divide 8x5y4 1 12x2y5 2 16x2y3 by 4x3y4, we proceed as follows: 52 8x5y4 12x2y5 216x2y3 8x5y4 1 12x2y5 2 16x2y3 5 3 41 3 4 3 4 1 4x y 4x y 4x y 4x3y4 5 2x2 1 3y 4 2 x xy To divide two polynomials, we can use long division. To illustrate, we consider the division 2x2 1 11x 2 30 x17 which can be written in long division form as x 1 7q2x2 1 11x 2 30 The binomial x 1 7 is called the divisor, and the trinomial 2x2 1 11x 2 30 is called the dividend. The final answer, called the quotient, will appear above the long division symbol. We begin the division by asking “What monomial, when multiplied by x, gives 2x2?” Because x ? 2x 5 2x2, the answer is 2x. We place 2x in the quotient, multiply each term of the divisor by 2x, subtract, and bring down the –30. 2x x 1 7q2x 1 11x 2 30 2x2 1 14x 2 3x 2 30 2 We continue the division by asking “What monomial, when multiplied by x, gives 23x?” We place the answer, 23, in the quotient, multiply each term of the divisor by 23, and subtract. This time, there is no number to bring down. 2x 2 3 x 1 7q2x 1 11x 2 30 2x2 1 14x 2 3x 2 30 2 3x 2 21 29 2 Because the degree of the remainder, 29, is less than the degree of the divisor, the division process stops, and we can express the result in the form quotient 1 remainder divisor © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A48 Section .4 Polynomials A49 Thus, 2x2 1 11x 2 30 29 5 2x 2 3 1 x17 x17 EXAMPLE 9 Using Long Division to Divide Polynomials Divide 6x3 2 11 by 2x 1 2. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Solution We set up the division, leaving spaces for the missing powers of x in the dividend. Comment 2x 1 2q6x3 2 11 In Example 9, we could write the missing powers of x using coefficients of 0. 2x 1 2q6x3 1 0x2 1 0x 2 11 The division process continues as usual, with the following results: 3x2 2 3x 1 3 2x 1 2q6x 2 11 3 2 6x 1 6x 2 6x2 2 6x2 2 6x 1 6x 2 11 1 6x 1 6 2 17 3 6x 2 11 217 Thus, . 5 3x2 2 3x 1 3 1 2x 1 2 2x 1 2 3 Self Check 9 Divide: 3x 1 1q9x2 2 1. Now Try Exercise 113. EXAMPLE 10 Using Long Division to Divide Polynomials Divide 23x3 2 3 1 x5 1 4x2 2 x4 by x2 2 3. Solution The division process works best when the terms in the divisor and dividend are written with their exponents in descending order. x3 2 x2 1 1 x2 2 3qx5 2 x4 2 3x3 1 4x2 2 3 x5 2 3x3 2 x4 1 4x2 4 2x 1 3x2 x2 2 3 x2 2 3 0 Thus, 23x3 2 3 1 x5 1 4x2 2 x4 5 x3 2 x2 1 1. x2 2 3 Self Check 10 Divide: x2 1 1q3x2 2 x 1 1 2 2x3 1 3x4. Not For Sale Now Try Exercise 115. Appendix A50 Not For Sale A Review of Basic Algebra Self Check Answers1. 7x2 2 2x 1 2 2. x2 1 8x 2 12 3. 8b2 2 7a2b 4. a. x2 1 4x 1 4 b. x2 2 6x 1 9 c. x2 2 16 5. 2x2 2 x"3 2 3 x 1 4"x 1 4 1 8. x24 "4 1 h 1 2 x11 2 9. 3x 2 1 10. 3x 2 2x 1 2 x 11 6. 2x3 1 7x2 1 5x 2 2 7. Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. A is a real number or the product of a real number and one or more . 2. The of a monomial is the sum of the exponents of its . 3. A is a polynomial with three terms. 4. A is a polynomial with two terms. 5. A monomial is a polynomial with term. 6. The constant 0 is called the polynomial. 7. Terms with the same variables with the same exponents are called terms. 8. The of a polynomial is the same as the degree of its term of highest degree. 9. To combine like terms, we add their and keep the same and the same exponents. 10. The conjugate of 3"x 1 2 is . Determine whether the given expression is a polynomial. If so, tell whether it is a monomial, a binomial, or a trinomial, and give its degree. 11. x2 1 3x 1 4 12. 5xy 2 x3 1/2 3 13. x 1 y 14. x23 2 5y22 2 3 15. 4x 2 "5x 2 3 16. x y 17. "15 x 5 18. 1 1 5 x 5 19. 0 20. 3y3 2 4y2 1 2y 1 2 Practice Perform the operations and simplify. 21. 1x3 2 3x22 1 15x3 2 8x2 22. 12x4 2 5x32 1 17x3 2 x4 1 2x2 23. 1y5 1 2y3 1 72 2 1y5 2 2y3 2 72 24. 13t7 2 7t3 1 32 2 17t7 2 3t3 1 72 25. 2 1x 1 3x 2 12 2 3 1x 1 2x 2 42 1 4 26. 5 1x3 2 8x 1 32 1 2 13x2 1 5x2 2 7 2 2 27. 8 1t2 2 2t 1 52 1 4 1t2 2 3t 1 22 2 6 12t2 2 82 28. 23 1x3 2 x2 1 2 1x2 1 x2 1 3 1x3 2 2x2 29. y 1y2 2 12 2 y2 1y 1 22 2 y 12y 2 22 30. 24a2 1a 1 12 1 3a 1a2 2 42 2 a2 1a 1 22 31. xy 1x 2 4y2 2 y 1x2 1 3xy2 1 xy 12x 1 3y2 32. 3mn 1m 1 2n2 2 6m 13mn 1 12 2 2n 14mn 2 12 33. 2x2y3 14xy42 34. 215a3b 122a2b32 mn b 12 3r2s3 2r2s 15rs2 36. 2 a ba b 5 3 2 35. 23m2n 12mn22 a2 37. 24rs 1r2 1 s22 38. 6u2v 12uv2 2 y2 39. 6ab2c 12ac 1 3bc2 2 4ab2c2 mn2 14mn 2 6m2 2 82 2 41. 1a 1 22 1a 1 22 42. 1y 2 52 1y 2 52 45. 1x 1 42 1x 2 42 46. 1z 1 72 1z 2 72 40. 2 43. 1a 2 62 2 44. 1t 1 92 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Exercises A.4 Section 47. 1x 2 32 1x 1 52 49. 1u 1 22 13u 2 22 51. 15x 2 12 12x 1 32 91. "2 2 "3 52. 14x 2 12 12x 2 72 93. "x 2 "y 50. 14x 1 12 12x 2 32 53. 13a 2 2b2 54. 14a 1 5b2 14a 2 5b2 57. 12y 2 4x2 13y 2 2x2 58. 122x 1 3y2 13x 1 y2 2 55. 13m 1 4n2 13m 2 4n2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 48. 1z 1 42 1z 2 62 59. 19x 2 y2 1x2 2 3y2 61. 15z 1 2t2 1z2 2 t2 63. Q"5 1 3xRQ2 2 "5xR 65. 13x 2 12 3 67. 13x 1 12 12x2 1 4x 2 32 68. 12x 2 52 1x2 2 3x 1 22 56. 14r 1 3s2 2 60. 18a2 1 b2 1a 1 2b2 1 2 "3 "x 1 "y 95. "2 1 1 2 97. y 2 "3 66. 12x 2 32 3 99. "x 1 3 2 "x 3 69. 13x 1 2y2 12x2 2 3xy 1 4y22 70. 14r 2 3s2 12r2 1 4rs 2 2s22 Multiply the expressions as you would multiply polynomials. 71. 2yn 13yn 1 y2n2 92. "3 2 "2 94. "2x 1 y 96. "x 2 3 3 1 1 "2 "2x 2 y Rationalize each numerator. 62. 1y 2 2x22 1x2 1 3y2 64. Q"2 1 xRQ3 1 "2xR .4 Polynomials A51 y 1 "3 98. "a 2 "b 100. "2 1 h 2 "2 h "a 1 "b Perform each division and write all answers without using negative exponents. 36a2b3 245r2s5t3 101. 102. 6 18ab 27r6s2t8 103. 16x6y4z9 224x9y6z0 104. 32m6n4p2 26m6n7p2 72. 3a2n 12an 1 3an212 105. 5x3y2 1 15x3y4 10x2y3 106. 9m4n9 2 6m3n4 12m3n3 75. 1xn 1 32 1xn 2 42 76. 1an 2 52 1an 2 32 107. 73. 25x2nyn 12x2ny2n 1 3x22nyn2 74. 22a3nb2n 15a23nb 2 ab22n2 24x5y7 2 36x2y5 1 12xy 60x5y4 3 4 2 4 2 3 9a b 1 27a b 2 18a b 108. 18a2b7 77. 12rn 2 72 13rn 2 22 78. 14zn 1 32 13zn 1 12 Perform each division. If there is a nonzero remainder, write the answer in quotient 1 remainder divisor form. 2 109. x 1 3q3x 1 11x 1 6 110. 3x 1 2q3x2 1 11x 1 6 79. x1/2 1x1/2y 1 xy1/22 80. ab1/2 1a1/2b1/2 1 b1/22 81. 1a1/2 1 b1/22 1a1/2 2 b1/22 82. 1x3/2 1 y1/22 2 Rationalize each denominator. 2 1 84. 83. "3 2 1 "5 1 2 3x 14y 85. 86. "7 1 2 "2 2 3 x y 87. 88. x 2 "3 2y 1 "7 89. y 1 "2 y 2 "2 111. 2x 2 5q2x2 2 19x 1 37 112. x 2 7q2x2 2 19x 1 35 2x3 1 1 113. x21 2x3 2 9x2 1 13x 2 20 114. 2x 2 7 115. x2 1 x 2 1qx3 2 2x2 2 4x 1 3 Not For Sale 90. x 2 "3 x 1 "3 116. x2 2 3qx3 2 2x2 2 4x 1 5 117. x5 2 2x3 2 3x2 1 9 x3 2 2 Not For Sale A Review of Basic Algebra x5 2 2x3 2 3x2 1 9 x3 2 3 x5 2 32 119. x22 x4 2 1 120. x11 126. Travel Complete the following table, which shows the rate (mph), time traveled (hr), and distance traveled (mi) by a family on vacation. 121. 11x 2 10 1 6x2 q36x4 2 121x2 1 120 1 72x3 2 142x Discovery and Writing 118. 122. x 1 6x2 2 12q2121x2 1 72x3 2 142x 1 120 1 36x4 Applications 123. Geometry Find an expression that represents the area of the brick wall. r ? 3x 1 4 5 t d 3x2 1 19x 1 20 127. Show that a trinomial can be squared by using the formula 1a 1 b 1 c2 2 5 a2 1 b2 1 c2 1 2ab 1 2bc 1 2ac. 128. Show that 1a 1 b 1 c 1 d2 2 5 a2 1 b2 1 c2 1 d 2 1 2ab 1 2ac 1 2ad 1 2bc 1 2bd 1 2cd . 1 29. Explain the FOIL method. 130. Explain how to rationalize the numerator of 131. Explain why 1a 1 b2 2 2 a2 1 b2. "x 1 2 . x 132. Explain why "a2 1 b2 2 "a2 1 "b2. (x – 2) ft Review (x + 5) ft 124. Geometry The area of the triangle shown in the illustration is represented as 1x2 1 3x 2 402 square feet. Find an expression that represents its height. Height (x + 8) ft 125. Gift Boxes The corners of a 12 in.-by-12 in. piece of cardboard are folded inward and glued to make a box. Write a polynomial that represents the volume of the resulting box. Crease here and fold inward x x x x 12 in. x x x x 12 in. Simplify each expression. Assume that all variables represent positive numbers. 8 22/3 133. 93/2 134. a b 125 625x4 3/4 135. a 136. "80x4 b 16y8 3 3 137. "16ab4 2 b"54ab 4 4 138. x" 1,280x 1 " 80x5 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Appendix A52 Section .5 Factoring Polynomials A53 A.5 Factoring Polynomials 1.Factor 2.Factor 3.Factor 4.Factor 5.Factor 6.Factor 7.Factor © Peter Coombs/Alamy © Cengage Learning. All rights reserved. No distribution allowed without express authorization. In this section, we will learn to out a common monomial. by grouping. the difference of two squares. trinomials. trinomials by grouping. the sum and difference of two cubes. miscellaneous polynomials. The television network series CSI: Crime Scene Investigation and its spinoffs, CSI: NY and CSI: Miami, are favorite television shows of many people. Their fans enjoy watching a team of forensic scientists uncover the circumstances that led to an unusual death or crime. These mysteries are captivating because many viewers are interested in the field of forensic science. In this section, we will investigate a mathematics mystery. We will be given a polynomial and asked to unveil or uncover the two or more polynomials that were multiplied together to obtain the given polynomial. The process that we will use to solve this mystery is called factoring. When two or more polynomials are multiplied together, each one is called a factor of the resulting product. For example, the factors of 7 1x 1 22 1x 1 32 are 7, x 1 2, and x 1 3 The process of writing a polynomial as the product of several factors is called factoring. In this section, we will discuss factoring where the coefficients of the polynomial and the polynomial factors are integers. If a polynomial cannot be factored by using integers only, we call it a prime polynomial. 1. Factor Out a Common Monomial The simplest type of factoring occurs when we see a common monomial factor in each term of the polynomial. In this case, our strategy is to use the Distributive Property and factor out the greatest common factor. EXAMPLE 1 Factoring by Removing a Common Monomial Factor: 3xy2 1 6x. SOLUTION We note that each term contains a greatest common factor of 3x: 3xy2 1 6x 5 3x 1y22 1 3x 122 We can then use the Distributive Property to factor out the common factor of 3x: 3xy2 1 6x 5 3x 1y2 1 22 We can check by multiplying: 3x 1y2 1 22 5 3xy2 1 6x. Self Check 1 Factor: 4a2 2 8ab. Now Try Exercise 11. Not For Sale A54 Appendix Not For Sale A Review of Basic Algebra EXAMPLE 2 Factoring by Removing a Common Monomial Factor: x2y2z2 2 xyz. SOLUTION We factor out the greatest common factor of xyz: x2y2z2 2 xyz 5 xyz 1xyz2 2 xyz 112 5 xyz 1xyz 2 12 We can check by multiplying. 2 2 2 Self Check 2 Factor: a2b2c2 1 a3b3c3. Now Try Exercise 13. 2. Factor by Grouping A strategy that is often helpful when factoring a polynomial with four or more terms is called grouping. Terms with common factors are grouped together and then their greatest common factors are factored out using the Distributive Property. EXAMPLE 3 Factoring by Grouping Factor: ax 1 bx 1 a 1 b. SOLUTION Although there is no factor common to all four terms, we can factor x out of the first two terms and write the expression as ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2 We can now factor out the common factor of a 1 b. ax 1 bx 1 a 1 b 5 x 1a 1 b2 1 1a 1 b2 5 x 1a 1 b2 1 1 1a 1 b2 5 1a 1 b2 1x 1 12 We can check by multiplying. Self Check 3 Factor: x2 1 xy 1 2x 1 2y. Now Try Exercise 17. 3. Factor the Difference of Two Squares A binomial that is the difference of the squares of two quantities factors easily. The strategy used is to write the polynomial as the product of two factors. The first factor is the sum of the quantities and the other factor is the difference of the quantities. EXAMPLE 4 Factoring the Difference of Two Squares Factor: 49x2 2 4. SOLUTION We observe that each term is a perfect square: 49x2 2 4 5 17x2 2 2 22 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. The last term in the expression x y z 2 xyz has an understood coefficient of 1. When the xyz is factored out, the 1 must be written. Section .5 Factoring Polynomials A55 The difference of the squares of two quantities is the product of two factors. One is the sum of the quantities, and the other is the difference of the quantities. Thus, 49x2 2 4 factors as 49x2 2 4 5 17x2 2 2 22 5 17x 1 22 17x 2 22 We can check by multiplying. Self Check 4 Factor: 9a2 2 16b2. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Now Try Exercise 21. Example 4 suggests a formula for factoring the difference of two squares. Factoring the Difference of Two Squares x2 2 y2 5 1x 1 y2 1x 2 y2 EXAMPLE 5 Factoring the Difference of Two Squares Twice in One Problem Factor: 16m4 2 n4. SOLUTION The binomial 16m4 2 n4 can be factored as the difference of two squares: Caution If you are limited to integer coefficients, the sum of two squares cannot be factored. For example, x2 1 y2 is a prime polynomial. 16m4 2 n4 5 14m22 2 2 1n22 2 5 14m2 1 n22 14m2 2 n22 The first factor is the sum of two squares and is prime. The second factor is a difference of two squares and can be factored: 16m4 2 n4 5 14m2 1 n22 3 12m2 2 2 n2 4 5 14m2 1 n22 12m 1 n2 12m 2 n2 We can check by multiplying. Self Check 5 Factor: a4 2 81b4. Now Try Exercise 23. EXAMPLE 6Removing a Common Factor and Factoring the Difference of Two Squares Factor: 18t2 2 32. SOLUTION We begin by factoring out the common monomial factor of 2. 18t2 2 32 5 2 19t2 2 162 Since 9t2 2 16 is the difference of two squares, it can be factored. 18t2 2 32 5 2 19t2 2 162 5 2 13t 1 42 13t 2 42 Self Check 6 Factor: 23x2 1 12. We can check by multiplying. Not For Sale Now Try Exercise 61. A56 Appendix Not For Sale A Review of Basic Algebra 4. Factor Trinomials Trinomials that are squares of binomials can be factored by using the following formulas. Factoring Trinomial Squares (1) x2 1 2xy 1 y2 5 1x 1 y2 1x 1 y2 5 1x 1 y2 2 For example, to factor a2 2 6a 1 9, we note that it can be written in the form a2 2 2 13a2 1 32 x 5 a and y 5 3 which matches the left side of Equation 2 above. Thus, a2 2 6a 1 9 5 a2 2 2 13a2 1 32 5 1a 2 32 1a 2 32 5 1a 2 32 2 We can check by multiplying. Factoring trinomials that are not squares of binomials often requires some guesswork. If a trinomial with no common factors is factorable, it will factor into the product of two binomials. EXAMPLE 7 Factoring a Trinomial with Leading Coefficient of 1 Factor: x2 1 3x 2 10. SOLUTION To factor x2 1 3x 2 10, we must find two binomials x 1 a and x 1 b such that x2 1 3x 2 10 5 1x 1 a2 1x 1 b2 where the product of a and b is –10 and the sum of a and b is 3. ab 5 210 and a 1 b 5 3 To find such numbers, we list the possible factorizations of –10: 10 1212 5 1222 210 112 25 122 Only in the factorization 5 1222 do the factors have a sum of 3. Thus, a 5 5 and b 5 22, and (3) x2 1 3x 2 10 5 1x 1 a2 1x 1 b2 x2 1 3x 2 10 5 1x 1 52 1x 2 22 We can check by multiplying. Because of the Commutative Property of Multiplication, the order of the factors in Equation 3 is not important. Equation 3 can also be written as x2 1 3x 2 10 5 1x 2 22 1x 1 52 Self Check 7 Factor: p2 2 5p 2 6. Now Try Exercise 29. EXAMPLE 8 Factoring a Trinomial with Leading Coefficient not 1 Factor: 2x2 2 x 2 6. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. (2) x2 2 2xy 1 y2 5 1x 2 y2 1x 2 y2 5 1x 2 y2 2 Section .5 Factoring Polynomials A57 SOLUTION Since the first term is 2x2, the first terms of the binomial factors must be 2x and x: 2x2 2 x 2 6 5 12x 2 1x 2 The product of the last terms must be 26, and the sum of the products of the outer terms and the inner terms must be 2x. Since the only factorization of 26 that will cause this to happen is 3 1222 , we have 2x2 2 x 2 6 5 12x 1 32 1x 2 22 We can check by multiplying. Self Check 8 Factor: 6x2 2 x 2 2. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Now Try Exercise 33. It is not easy to give specific rules for factoring trinomials, because some guesswork is often necessary. However, the following hints are helpful. Strategy for Factoring a General Trinomial with Integer Coefficients 1. Write the trinomial in descending powers of one variable. 2. Factor out any greatest common factor, including 21 if that is necessary to make the coefficient of the first term positive. 3. When the sign of the first term of a trinomial is 1 and the sign of the third term is 1, the sign between the terms of each binomial factor is the same as the sign of the middle term of the trinomial. When the sign of the first term is 1 and the sign of the third term is 2, one of the signs between the terms of the binomial factors is 1 and the other is 2. 4. Try various combinations of first terms and last terms until you find one that works. If no possibilities work, the trinomial is prime. 5. Check the factorization by multiplication. EXAMPLE 9 Factoring a Trinomial Completely Factor: 10xy 1 24y2 2 6x2. SOLUTION We write the trinomial in descending powers of x and then factor out the common factor of 22. 10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2 5 22 13x2 2 5xy 2 12y22 Since the sign of the third term of 3x2 2 5xy 2 12y2 is 2, the signs between the binomial factors will be opposite. Since the first term is 3x2, the first terms of the binomial factors must be 3x and x: 22 13x2 2 5xy 2 12y22 5 22 13x 2 1x 2 The product of the last terms must be 212y2, and the sum of the outer terms and the inner terms must be 25xy. Of the many factorizations of 212y2, only 4y 123y2 leads to a middle term of 25xy. So we have 10xy 1 24y2 2 6x2 5 26x2 1 10xy 1 24y2 5 22 13x2 2 5xy 2 12y22 Not For Sale 5 22 13x 1 4y2 1x 2 3y2 We can check by multiplying. A58 Appendix Not For Sale A Review of Basic Algebra Self Check 9 Factor: 26x2 2 15xy 2 6y2. Now Try Exercise 69. 5. Factor Trinomials by Grouping 1. Find the product ac: 6 1262 5 236. This number is called the key number. 2. Find two factors of the key number 12362 whose sum is b 5 5. Two such numbers are 9 and 24. 9 1242 5 236 and 9 1 1242 5 5 3. Use the factors 9 and 24 as coefficients of two terms to be placed between 6x2 and 26. 6x2 1 5x 2 6 5 6x2 1 9x 2 4x 2 6 4. Factor by grouping: 6x2 1 9x 2 4x 2 6 5 3x 12x 1 32 2 2 12x 1 32 5 12x 1 32 13x 2 22 Factor out 2x 1 3. EXAMPLE 10 Factoring a Trinomial by Grouping Factor: 15x2 1 x 2 2. SOLUTION Since a 5 15 and c 5 22 in the trinomial, ac 5 230. We now find factors of 230 whose sum is b 5 1. Such factors are 6 and 25. We use these factors as coefficients of two terms to be placed between 15x2 and –2. 15x2 1 6x 2 5x 2 2 Finally, we factor by grouping. 3x 15x 1 22 2 15x 1 22 5 15x 1 22 13x 2 12 Self Check 10 Factor: 15a2 1 17a 2 4. Now Try Exercise 39. We can often factor polynomials with variable exponents. For example, if n is a natural number, a2n 2 5an 2 6 5 1an 1 12 1an 2 62 because 1an 1 12 1an 2 62 5 a2n 2 6an 1 an 2 6 5 a2n 2 5an 2 6 Combine like terms. 6. Factor the Sum and Difference of Two Cubes Two other types of factoring involve binomials that are the sum or the difference of two cubes. Like the difference of two squares, they can be factored by using a formula. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Another way of factoring trinomials involves factoring by grouping. This method can be used to factor trinomials of the form ax2 1 bx 1 c. For example, to factor 6x2 1 5x 2 6, we note that a 5 6, b 5 5, and c 5 26, and proceed as follows: Section Factoring the Sum and Difference of Two Cubes .5 Factoring Polynomials A59 x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22 x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22 EXAMPLE 11 Factoring the Sum of Two Cubes Factor: 27x6 1 64y3. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. SOLUTION We can write this expression as the sum of two cubes and factor it as follows: 27x6 1 64y3 5 13x22 3 1 14y2 3 5 13x2 1 4y2 3 13x22 2 2 13x22 14y2 1 14y2 2 4 5 13x2 1 4y2 19x4 2 12x2y 1 16y22 We can check by multiplying. Self Check 11 Factor: 8a3 1 1,000b6. Now Try Exercise 41. EXAMPLE 12 Factoring the Difference of Two Cubes Factor: x3 2 8. SOLUTION This binomial can be written as x3 2 23, which is the difference of two cubes. Substituting into the formula for the difference of two cubes gives x3 2 23 5 1x 2 22 1x2 1 2x 1 222 5 1x 2 22 1x2 1 2x 1 42 We can check by multiplying. Self Check 12 Factor: p3 2 64. Now Try Exercise 43. 7. Factor Miscellaneous Polynomials EXAMPLE 13 Factoring a Miscellaneous Polynomial Factor: x2 2 y2 1 6x 1 9. SOLUTION Here we will factor a trinomial and a difference of two squares. x2 2 y2 1 6x 1 9 5 x2 1 6x 1 9 2 y2 5 1x 1 32 2 2 y2 Use the Commutative Property to rearrange the terms. Factor x2 1 6x 1 9. 5 1x 1 3 1 y2 1x 1 3 2 y2 Factor the difference of two squares. We could try to factor this expression in another way. x2 2 y2 1 6x 1 9 5 1x 1 y2 1x 2 y2 1 3 12x 1 32 Factor x2 2 y2 and 6x 1 9. However, we are unable to finish the factorization. If grouping in one way doesn’t work, try various other ways. Not For Sale Self Check 13 Factor: a2 1 8a 2 b2 1 16. Now Try Exercise 97. A60 Appendix Not For Sale A Review of Basic Algebra EXAMPLE 14 Factoring a Miscellaneous Trinomial Factor: z4 2 3z2 1 1. SOLUTION This trinomial cannot be factored as the product of two binomials, because no combination will give a middle term of 23z2. However, if the middle term were 22z2, the trinomial would be a perfect square, and the factorization would be easy: z4 2 2z2 1 1 5 1z2 2 12 1z2 2 12 We can change the middle term in z4 2 3z2 1 1 to 22z2 by adding z2 to it. However, to make sure that adding z2 does not change the value of the trinomial, we must also subtract z2. We can then proceed as follows. z4 2 3z2 1 1 5 z4 2 3z2 1 z2 1 1 2 z2 Add and subtract z2. 5 z4 2 2z2 1 1 2 z2 Combine 23z2 and z2. 5 1z2 2 12 2 2 z2 Factor z4 2 2z2 1 1. 5 1z2 2 1 1 z2 1z2 2 1 2 z2 Factor the difference of two squares. In this type of problem, we will always try to add and subtract a perfect square in hopes of making a perfect-square trinomial that will lead to factoring a difference of two squares. Self Check 14 Factor: x4 1 3x2 1 4. Now Try Exercise 103. It is helpful to identify the problem type when we must factor polynomials that are given in random order. Factoring Strategy 1. Factor out all common monomial factors. 2. If an expression has two terms, check whether the problem type is a. The difference of two squares: x2 2 y2 5 1x 1 y2 1x 2 y2 b. The sum of two cubes: x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22 c. The difference of two cubes: x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22 3. If an expression has three terms, try to factor it as a trinomial. 4. If an expression has four or more terms, try factoring by grouping. 5. Continue until each individual factor is prime. 6. Check the results by multiplying. Self Check Answers 1. 4a 1a 2 2b2 2. a2b2c2 11 1 abc2 3. 1x 1 y2 1x 1 22 4. 13a 1 4b2 13a 2 4b2 5. 1a2 1 9b22 1a 1 3b2 1a 2 3b2 6. 23 1x 1 22 1x 2 22 7. 1p 2 62 1p 1 12 8. 13x 2 22 12x 1 12 9. 23 1x 1 2y2 12x 1 y2 10. 13a 1 42 15a 2 12 11. 8 1a 1 5b22 1a2 2 5ab2 1 25b42 12. 1p 2 42 1p2 1 4p 1 162 13. 1a 1 4 1 b2 1a 1 4 2 b2 14. 1x2 1 2 1 x2 1x2 1 2 2 x2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 5 1z2 2 12 2 Section .5 Factoring Polynomials A61 Exercises A.5 Getting Ready © Cengage Learning. All rights reserved. No distribution allowed without express authorization. You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. When polynomials are multiplied together, each polynomial is a of the product. 2. If a polynomial cannot be factored using coefficients, it is called a polynomial. Complete each factoring formula. 3. ax 1 bx 5 2 2 4. x 2 y 5 3 3. 12x2 2 xy 2 6y2 34. 8x2 2 10xy 2 3y2 In each expression, factor the trinomial by grouping. 35. x2 1 10x 1 21 36. x2 1 7x 1 10 37. x2 2 4x 2 12 38. x2 2 2x 2 63 39. 6p2 1 7p 2 3 40. 4q2 2 19q 1 12 In each expression, factor the sum of two cubes. 41. t3 1 343 42. r3 1 8s3 In each expression, factor the difference of two cubes. 5. x2 1 2xy 1 y2 5 4 3. 8z3 2 27 6. x2 2 2xy 1 y2 5 7. x3 1 y3 5 44. 125a3 2 64 8. x3 2 y3 5 Factor each expression completely. If an expression is prime, so indicate. Practice 45. 3a2bc 1 6ab2c 1 9abc2 In each expression, factor out the greatest common monomial. 9. 3x 2 6 2 3 1 1. 8x 1 4x 13. 7x2y2 1 14x3y2 10. 5y 2 15 12. 9y3 1 6y2 14. 25y2z 2 15yz2 In each expression, factor by grouping. 15. a 1x 1 y2 1 b 1x 1 y2 16. b 1x 2 y2 1 a 1x 2 y2 17. 4a 1 b 2 12a2 2 3ab 18. x2 1 4x 1 xy 1 4y In each expression, factor the difference of two squares. 20. 36z2 2 49 19. 4x2 2 9 21. 4 2 9r2 22. 16 2 49x2 23. 81x4 2 1 24. 81 2 x4 25. 1x 1 z2 2 2 25 26. 1x 2 y2 2 2 9 In each expression, factor the trinomial. 27. x2 1 8x 1 16 28. a2 2 12a 1 36 29. b2 2 10b 1 25 30. y2 1 14y 1 49 31. m2 1 4mn 1 4n2 32. r2 2 8rs 1 16s2 46. 5x3y3z3 1 25x2y2z2 2 125xyz 47. 3x3 1 3x2 2 x 2 1 48. 4x 1 6xy 2 9y 2 6 49. 2txy 1 2ctx 2 3ty 2 3ct 50. 2ax 1 4ay 2 bx 2 2by 51. ax 1 bx 1 ay 1 by 1 az 1 bz 5 2. 6x2y3 1 18xy 1 3x2y2 1 9x 53. x2 2 1y 2 z2 2 54. z2 2 1y 1 32 2 55. 1x 2 y2 2 2 1x 1 y2 2 56. 12a 1 32 2 2 12a 2 32 2 57. x4 2 y4 58. z4 2 81 59. 3x2 2 12 60. 3x3y 2 3xy 61. 18xy2 2 8x 62. 27x2 2 12 63. x2 2 2x 1 15 64. x2 1 x 1 2 65. 215 1 2a 1 24a2 66. 232 2 68x 1 9x2 67. 6x2 1 29xy 1 35y2 68. 10x2 2 17xy 1 6y2 69. 12p2 2 58pq 2 70q2 70. 3x2 2 6xy 2 9y2 71. 26m2 1 47mn 2 35n2 73. 26x3 1 23x2 1 35x 72. 214r2 2 11rs 1 15s2 Not For Sale 74. 2y3 2 y2 1 90y Not For Sale A Review of Basic Algebra 75. 6x4 2 11x3 2 35x2 76. 12x 1 17x2 2 7x3 77. x4 1 2x2 2 15 78. x4 2 x2 2 6 79. a2n 2 2an 2 3 80. a2n 1 6an 1 8 81. 6x2n 2 7xn 1 2 82. 9x2n 1 9xn 1 2 83. 4x2n 2 9y2n 84. 8x2n 2 2xn 2 3 2n 4n n 110. Movie Stunts The formula that gives the distance a stuntwoman is above the ground t seconds after she falls over the side of a 144-foot tall building is f 5 144 2 16t2. Factor the right side. 144 ft 2n 85. 10y 2 11y 2 6 86. 16y 2 25y 87. 2x3 1 2,000 88. 3y3 1 648 89. 1x 1 y2 3 2 64 90. 1x 2 y2 3 1 27 Discovery and Writing 111. Explain how to factor the difference of two squares. 112. Explain how to factor the difference of two cubes. 91. 64a6 2 y6 92. a6 1 b6 113. Explain how to factor a2 2 b2 1 a 1 b. 93. a3 2 b3 1 a 2 b 94. 1a2 2 y22 2 5 1a 1 y2 114. Explain how to factor x2 1 2x 1 1. 95. 64x6 1 y6 Factor the indicated monomial from the given expression. 96. z2 1 6z 1 9 2 225y2 115. 3x 1 2; 2 97. x2 2 6x 1 9 2 144y2 98. x2 1 2x 2 9y2 1 1 99. 1a 1 b2 2 2 3 1a 1 b2 2 10 100. 2 1a 1 b2 2 2 5 1a 1 b2 2 3 102. x6 2 13x4 1 36x2 103. x4 1 x2 1 1 104. x4 1 3x2 1 4 4 118. 3x2 2 2x 2 5; 3 119. a 1 b; a 120. a 2 b; b 1/2 101. x6 1 7x3 2 8 116. 5x 2 3; 5 117. x2 1 2x 1 4; 2 1/2 121. x 1 x ; x 122. x3/2 2 x1/2 ; x1/2 123. 2x 1 "2y; "2 124. "3a 2 3b; "3 125. ab3/2 2 a3/2b; ab 126. ab2 1 b; b21 2 105. x 1 7x 1 16 106. y4 1 2y2 1 9 107. 4a4 1 1 1 3a2 Factor each expression by grouping three terms and two terms. 108. x4 1 25 1 6x2 127. x2 1 x 2 6 1 xy 2 2y 109. Candy To find the amount of chocolate used in the outer coating of one of the malted-milk balls shown, we can find the volume V of the chocolate shell using the formula V 5 43pr31 2 43pr32. Factor the expression on the right side of the formula. 128. 2x2 1 5x 1 2 2 xy 2 2y 129. a4 1 2a3 1 a2 1 a 1 1 130. a4 1 a3 2 2a2 1 a 2 1 Review Outer radius r1 © Istockphoto.com/maureenpr Inner radius r2 131. Which natural number is neither prime nor composite? 132. Graph the interval 3 22,32 . 133. Simplify: 1x3x22 4. 1a32 3 1a22 4 134. Simplify: . 1a2a32 3 136. Simplify: "20x5. 135. a 137. Simplify: "20x 2 "125x. 138. Rationalize the denominator: 3x4x3 0 b 6x22x4 3 . !3 3 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Appendix A62 Section .6 RationalExpressions A63 A.6 Rational Expressions In this section, we will learn to © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 1.Define rational expressions. 2.Simplify rational expressions. 3.Multiple and divide rational expressions. 4.Add and subtract rational expressions. 5.Simplify complex fractions. Photo by Tim Boyle/Getty Images Abercrombie and Fitch (A&F) is a very successful American clothing company founded in 1892 by David Abercrombie and Ezra Fitch. Today the stores are popular shopping destinations for university students wanting to keep up with the latest styles and trends. Suppose that a clothing manufacturer finds that the cost in dollars of producing x fleece vintage shirts is given by the algebraic expression 13x 1 1,000. The average cost of producing each shirt could be obtained by dividing the production cost, 13x 1 1,000, by the number of shirts produced, x. The algebraic fraction 13x 1 1,000 x represents the average cost per shirt. We see that the average cost of producing 200 shirts would be $18. 13 12002 1 1,000 5 18 200 An understanding of algebraic fractions is important in solving many real-life problems. 1. Define Rational Expressions Caution If x and y are real numbers, the quotient xy 1y 2 02 is called a fraction. The number x is called the numerator, and the number y is called the denominator. Algebraic fractions are quotients of algebraic expressions. If the expressions are polynomials, the fraction is called a rational expression. The first two of the following algebraic fractions are rational expressions. The third is not, because the numerator and denominator are not polynomials. Not For Sale Remember that the denominator of a fraction cannot be zero. 5y2 1 2y 8ab2 2 16c3 x1/2 1 4x y2 2 3y 2 7 2x 1 3 x3/2 2 x1/2 A64 Appendix Not For Sale A Review of Basic Algebra We summarize some of the properties of fractions as follows: Properties of Fractions If a, b, c, and d are real numbers and no denominators are 0, then Equality of Fractions a c 5 if and only if ad 5 bc b d Fundamental Property of Fractions Multiplication and Division of Fractions a c ac a c a d ad ? 5 and 4 5 ? 5 b d bd b d b c bc Addition and Subtraction of Fractions a c a1c a c a2c and 1 5 2 5 b b b b b b The first two examples illustrate each of the previous properties of fractions. EXAMPLE 1Illustrating the Properties of Fractions Assume that no denominators are 0. 2a 4a 5 Because 2a 162 5 3 14a2 a. 3 6 b. 6xy 3 12xy2 5 Factor the numerator and denominator and divide out the common factors. 10xy 5 12xy2 5 Self Check 1 a. Is 3 5 3y 15z 5 ? 5 25 b. Simplify: 15a2b . 25ab2 Now Try Exercise 9. EXAMPLE 2Illustrating the Properties of Fractions Assume that no denominators are 0. a. c. 2r 3r 2r ? 3r ? 5 7s 5s 7s ? 5s 5 6r2 35s2 b. 2ab ab 2ab 1 ab 1 5 5xy 5xy 5xy 5 3ab 5xy d. 3mn 2pq 3mn 7mn 4 5 ? 4pq 7mn 4pq 2pq 5 21m2n2 8p2q2 6uv2 3uv2 6uv2 2 3uv2 2 2 2 5 7w 7w 7w2 5 3uv2 7w2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. ax a 5 bx b Section Self Check 2 Perform each operation: 3a 2a 2ab 2rs a. ? b. 4 5b 7b 3rs 4ab c. .6 RationalExpressions A65 5pq 3pq 1 3t 3t d. 5mn2 mn2 2 3w 3w Now Try Exercise 19. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. To add or subtract rational expressions with unlike denominators, we write each expression as an equivalent expression with a common denominator. We can then add or subtract the expressions. For example, 3x 2x 3x 172 2x 152 1 5 1 5 7 5 172 7 152 4a2 3a2 4a2 122 3a2 132 2 5 2 15 10 15 122 10 132 5 8a2 9a2 2 30 30 5 8a2 2 9a2 30 5 5 21x 10x 1 35 35 5 21x 1 10x 35 5 31x 35 2a2 30 a2 52 30 A rational expression is in lowest terms if all factors common to the numerator and the denominator have been removed. To simplify a rational expression means to write it in lowest terms. 2. Simplify Rational Expressions To simplify rational expressions, we use the Fundamental Property of Fractions. This enables us to divide out all factors that are common to the numerator and the denominator. EXAMPLE 3Simplifying a Rational Expression Simplify: x2 2 9 1x 2 0, 32 . x2 2 3x SOLUTION We factor the difference of two squares in the numerator, factor out x in the denominator, and divide out the common factor of x 2 3. 1x 1 32 1x 2 32 x2 2 9 5 2 x 2 3x x 1x 2 32 5 Self Check 3 Simplify: x23 51 x23 x13 x a2 2 4a 1a 2 4,232 . a2 2 a 2 12 Not For Sale Now Try Exercise 23. We will encounter the following properties of fractions in the next examples. A66 Appendix Not For Sale A Review of Basic Algebra Properties of Fractions If a and b represent real numbers and there are no divisions by 0, then • a 5 a 1 • • a 2a a 2a 5 52 52 b 2b 2b b a a 2a 2a • 2 5 5 52 b 2b b 2b a 51 a Simplify: x2 2 2xy 1 y2 1x 2 y2 . y2x SOLUTION We factor the trinomial in the numerator, factor 21 from the denominator, and divide out the common factor of x 2 y. 1x 2 y2 1x 2 y2 x2 2 2xy 1 y2 5 y2x 21 1x 2 y2 x2y 5 21 x2y 52 1 5 2 1x 2 y2 Self Check 4 Simplify: x2y x2y 51 a2 2 ab 2 2b2 12b 2 a 2 02 . 2b 2 a Now Try Exercise 25. EXAMPLE 5Simplifying a Rational Expression Simplify: x2 2 3x 1 2 1x 2 2,212 . x2 2 x 2 2 SOLUTION We factor the numerator and denominator and divide out the common factor of x 2 2. 1x 2 12 1x 2 22 x2 2 3x 1 2 5 2 1x 1 12 1x 2 22 x 2x22 5 Self Check 5 Simplify: x22 51 x22 x21 x11 a2 1 3a 2 4 1a 2 1,232 . a2 1 2a 2 3 Now Try Exercise 27. 3. Multiply and Divide Rational Expressions EXAMPLE 6 Multiplying Rational Expressions Multiply: x2 2 x 2 2 x2 1 2x 2 3 ? 1x 2 1, 21, 22 . x2 2 1 x22 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. EXAMPLE 4Simplifying a Rational Expression Section .6 RationalExpressions A67 SOLUTION To multiply the rational expressions, we multiply the numerators, multiply the denominators, and divide out the common factors. 1x2 2 x 2 22 1x2 1 2x 2 32 x2 2 x 2 2 x2 1 2x 2 3 ? 5 2 1x2 2 12 1x 2 22 x 21 x22 5 1x 2 22 1x 1 12 1x 2 12 1x 1 32 1x 1 12 1x 2 12 1x 2 22 x22 x11 x21 5 1, 5 1, 51 x22 x11 x21 5x13 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Self Check 6 Simplify: x2 2 9 x 2 1 ? 1x 2 0, 1, 32 . x2 2 x x2 2 3x Now Try Exercise 33. EXAMPLE 7Dividing Rational Expressions Divide: x2 2 2x 2 3 x2 1 2x 2 15 4 2 1x 2 2, 22, 3, 252 . 2 x 24 x 1 3x 2 10 SOLUTION To divide the rational expressions, we multiply by the reciprocal of the second rational expression. We then simplify by factoring the numerator and denominator and dividing out the common factors. x2 2 2x 2 3 x2 1 2x 2 15 x2 2 2x 2 3 x2 1 3x 2 10 4 5 ? 2 x2 2 4 x2 1 3x 2 10 x2 2 4 x 1 2x 2 15 5 5 5 Self Check 7 Simplify: 1x2 2 2x 2 32 1x2 1 3x 2 102 1x2 2 42 1x2 1 2x 2 152 1x 2 32 1x 1 12 1x 2 22 1x 1 52 1x 1 22 1x 2 22 1x 1 52 1x 2 32 x23 x22 x15 5 1, 5 1, 51 x23 x22 x15 x11 x12 a2 2 a a2 2 2a 1a 2 0, 2, 222 . 4 2 a12 a 24 Now Try Exercise 39. EXAMPLE 8 Using Multiplication and Division to Simplify a Rational Expression Simplify: 2x2 2 5x 2 3 3x2 1 2x 2 1 2x2 1 x 1 1 ? 2 4 ax 2 , 21, 3, 0, 2 b. 3x 2 1 x 2 2x 2 3 3x 3 2 SOLUTION We can change the division to a multiplication, factor, and simplify. 2x2 2 5x 2 3 3x2 1 2x 2 1 2x2 1 x ? 2 4 3x 2 1 x 2 2x 2 3 3x 2x2 2 5x 2 3 3x2 1 2x 2 1 3x ? 2 ? 2 3x 2 1 x 2 2x 2 3 2x 1 x 2 12x 2 5x 2 32 13x2 1 2x 2 12 13x2 5 13x 2 12 1x2 2 2x 2 32 12x2 1 x2 53 5 5 1x 2 32 12x 1 12 13x 2 12 1x 1 12 3x 13x 2 12 1x 1 12 1x 2 32 x 12x 1 12 x23 2x 1 1 3x 2 1 x11 x 5 1, 5 1, 5 1, 5 1, 5 1 x23 2x 1 1 3x 2 1 x11 x Not For Sale A68 Appendix Not For Sale A Review of Basic Algebra Self Check 8 Simplify: x2 2 25 x2 2 5x x2 1 2x 4 2 ? 1x 2 0, 2, 5, 252 . x22 x 2 2x x2 1 5x Now Try Exercise 45. 4. Add and Subtract Rational Expressions EXAMPLE 9Adding Rational Expression with Like Deonominators Add: SOLUTION 2x 1 5 3x 1 20 1 1x 2 252 . x15 x15 2x 1 5 3x 1 20 5x 1 25 1 5 x15 x15 x15 5 1x 1 52 Factor out 5 and divide out the common factor of x 1 5. 1 1x 1 52 5 55 Self Check 9 Add: Add the numerators and keep the common denominator. 3x 2 2 x26 1 1x 2 22 . x22 x22 Now Try Exercise 49. To add (or subtract) rational expressions with unlike denominators, we must find a common denominator, called the least (or lowest) common denominator (LCD). Suppose the unlike denominators of three rational expressions are 12, 20, and 35. To find the LCD, we first find the prime factorization of each number. 12 5 4 ? 3 20 5 4 ? 5 35 5 5 ? 7 5 22 ? 3 5 22 ? 5 Because the LCD is the smallest number that can be divided by 12, 20, and 35, it must contain factors of 22, 3, 5, and 7. Thus, the LCD 5 22 ? 3 ? 5 ? 7 5 420 Comment Remember to find the least common denominator, use each factor the greatest number of times that it occurs. That is, 420 is the smallest number that can be divided without remainder by 12, 20, and 35. When finding an LCD, we always factor each denominator and then create the LCD by using each factor the greatest number of times that it appears in any one denominator. The product of these factors is the LCD. This rule also applies if the unlike denominators of the rational expressions contain variables. Suppose the unlike denominators are x2 1x 2 52 and x 1x 2 52 3. To find the LCD, we use x2 and 1x 2 52 3. Thus, the LCD is the product x2 1x 2 52 3 . EXAMPLE 10Adding Rational Expressions with Unlike Denominators Add: 1 2 1 2 1x 2 2, 222 . x2 2 4 x 2 4x 1 4 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. To add (or subtract) rational expressions with like denominators, we add (or subtract) the numerators and keep the common denominator. Section .6 RationalExpressions A69 SOLUTION We factor each denominator and find the LCD. x2 2 4 5 1x 1 22 1x 2 22 x2 2 4x 1 4 5 1x 2 22 1x 2 22 5 1x 2 22 2 The LCD is 1x 1 22 1x 2 22 2. We then write each rational expression with its denominator in factored form, convert each rational expression into an equivalent expression with a denominator of 1x 1 22 1x 2 22 2, add the expressions, and simplify. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 1 2 1 2 1 2 5 1 1x 1 22 1x 2 22 1x 2 22 1x 2 22 x2 2 4 x 2 4x 1 4 5 5 Comment 5 Always attempt to simplify the final result. In this case, the final fraction is already in lowest terms. 5 Self Check 10 Add: 1 1x 2 22 2 1x 1 22 1 1x 1 22 1x 2 22 1x 2 22 1x 2 22 1x 2 22 1x 1 22 x22 x12 5 1, 51 x22 x12 1 1x 2 22 1 2 1x 1 22 1x 1 22 1x 2 22 1x 2 22 x 2 2 1 2x 1 4 1x 1 22 1x 2 22 1x 2 22 3x 1 2 1x 1 22 1x 2 22 2 3 1 1 2 1x 2 3,232 . x 2 6x 1 9 x 29 2 Now Try Exercise 59. EXAMPLE 11 Combining and Simplifying Rational Expressions with Unlike Denominators Simplify: x22 x13 3 1x 2 1, 21, 222 . 2 2 1 2 2 x 21 x 1 3x 1 2 x 1x22 SOLUTION We factor the denominators to find the LCD. x2 2 1 5 1x 1 12 1x 2 12 x2 1 3x 1 2 5 1x 1 22 1x 1 12 x2 1 x 2 2 5 1x 1 22 1x 2 12 The LCD is 1x 1 12 1x 1 22 1x 2 12 . We now write each rational expression as an equivalent expression with this LCD, and proceed as follows: x22 x13 3 2 2 1 2 2 x 21 x 1 3x 1 2 x 1x22 x22 x13 3 5 2 1 1x 1 12 1x 2 12 1x 1 12 1x 1 22 1x 2 12 1x 1 22 1x 2 22 1x 1 22 1x 1 32 1x 2 12 3 1x 1 12 5 2 1 1x 1 12 1x 2 12 1x 1 22 1x 1 12 1x 1 22 1x 2 12 1x 2 12 1x 1 22 1x 1 12 1x2 2 42 2 1x2 1 2x 2 32 1 13x 1 32 5 1x 1 12 1x 1 22 1x 2 12 5 5 x12 x21 x11 5 1, 5 1, 51 x12 x21 x11 x2 2 4 2 x2 2 2x 1 3 1 3x 1 3 1x 1 12 1x 1 22 1x 2 12 x12 1x 1 12 1x 1 22 1x 2 12 1 5 1x 1 12 1x 2 12 Not For Sale Divide out the common factor of x 1 2. x12 51 x12 A70 Appendix Not For Sale A Review of Basic Algebra Self Check 11 Simplify: 4y 2 2 1 2 1y 2 1, 212 . y2 2 1 y11 Now Try Exercise 61. 5. Simplify Complex Fractions Strategies for Simplifying Complex Fractions Method 1: Multiply the Complex Fraction by 1 • Determine the LCD of all fractions in the complex fraction. • Multiply both numerator and denominator of the complex fraction by the LCD. Note that when we multiply by LCD LCD we are multiplying by 1. Method 2: Simplify the Numerator and Denominator and then Divide • Simplify the numerator and denominator so that both are single fractions. • Perform the division by multiplying the numerator by the reciprocal of the denominator. EXAMPLE 12Simplifying Complex Fractions 1 1 1 x y Simplify: 1x, y 2 02 . x y Method 1: We note that the LCD of the three fractions in the complex fraction is xy. So we multiply the numerator and denominator of the complex fraction by xy and simplify: 1 1 1 xy xy 1 1 xya 1 b 1 x y x y x y y1x 5 5 5 x x xyx x2 xya b y y y Method 2: We combine the fractions in the numerator of the complex fraction to obtain a single fraction over a single fraction. 1 1 1y2 1 1x2 y1x 1 1 1 x y x 1y2 y 1x2 xy 5 5 x x x y y y Then we use the fact that any fraction indicates a division: y1x 1y 1 x2 y xy y1x x y1x y y1x 5 4 5 ? 5 5 x xy y xy x xyx x2 y © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A complex fraction is a fraction that has a fraction in its numerator or a fraction in its denominator. There are two methods generally used to simplify complex fractions. These are stated here for you. Section .6 RationalExpressions A71 1 1 2 x y Self Check 12 Simplify: 1x, y 2 02 . 1 1 1 x y Now Try Exercise 81. 3a 6a2 4a2b2 8pq 4mn2 2. a. d. 2 b. 2 2 c. 5b 35b 3r s 3t 3w a a14 x13 3. 4. 2 1a 1 b2 5. 6. 7. a 2 1 a13 a13 x2 4x 1 6 2y y2x 8. x 1 2 9. 4 10. 11. 12. 1x 1 32 1x 2 32 2 y21 y1x Self Check Answers 1. a. no b. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Exercises A.6 Getting Ready Perform the operations and simplify, whenever possible. Assume that no denominators are 0. 4x 2 25y 4 15. ? 16. ? 7 5a 2z y2 8m 3m 15p 25p 17. 4 18. 4 5n 10n 8q 16q2 3z 2z 7a 3a 19. 1 20. 2 5c 5c 4b 4b You should be able to complete these vocabulary and concept statements before you proceed to the practice exercises. Fill in the blanks. 1. In the fraction ab, a is called the . 2. In the fraction ab , b is called the . a c 3. b 5 d if and only if . 4. The denominator of a fraction can never be Complete each formula. a c 5. ? 5 b d 7. a c 1 5 b b a c 6. 4 5 b d 8. a c 2 5 b b . 21. 9. 8x 16x , 3y 6y 10. 1 1. 25xyz 50a2bc , 12ab2c 24xyz 12. Practice 2 3x 12y , 4y2 16x2 15rs2 37.5a3 , 4rs2 10a3 Simplify each rational expression. Assume that no denominators are 0. 7a2b 35p3q2 13. 14. 2 21ab 49p4q 22. 8rst2 7rst2 4 2 1 15m t 15m4t2 Simplify each fraction. Assume that no denominators are 0. 2x 2 4 x2 2 16 23. 2 24. 2 x 24 x 2 8x 1 16 25. 4 2 x2 x2 2 5x 1 6 26. 25 2 x2 x2 1 10x 1 25 27. 6x3 1 x2 2 12x 4x3 1 4x2 2 3x 28. 6x4 2 5x3 2 6x2 2x3 2 7x2 2 15x 29. x3 2 8 x 1 ax 2 2x 2 2a 30. xy 1 2x 1 3y 1 6 x3 1 27 Determine whether the fractions are equal. Assume that no denominators are 0. 2 15x2y x2y 2 3 2 7a b 7a2b3 2 Perform the operations and simplify, whenever possible. Assume that no denominators are 0. x2 2 1 x2 31. ? 2 x x 1 2x 1 1 Not For Sale Not For Sale A Review of Basic Algebra 32. y2 2 2y 1 1 y12 ? 2 y y 1y22 54. 3 3. 3x2 1 7x 1 2 x2 2 x ? 2 x2 1 2x 3x 1 x 55. 34. x2 1 x 2x2 1 x 2 3 ? 2 2x 1 3x x2 2 1 35. x2 1 x x2 2 1 ? x21 x12 x2 1 5x 1 6 x 1 2 36. 2 ? x 1 6x 1 9 x2 2 4 2x2 1 32 x2 1 16 37. 4 8 2 38. x2 1 x 2 6 x2 2 4 4 2 2 x 2 6x 1 9 x 29 39. z2 1 z 2 20 z2 2 25 4 2 z 24 z25 40. ax 1 bx 1 a 1 b x2 2 1 4 a2 1 2ab 1 b2 x2 2 2x 1 1 41. 3x2 1 5x 2 2 6x2 1 13x 2 5 4 x3 1 2x2 2x3 1 5x2 42. x2 1 13x 1 12 2x2 2 x 2 3 4 8x2 2 6x 2 5 8x2 2 14x 1 5 43. x2 1 7x 1 12 x2 2 3x 2 10 x3 2 4x2 1 3x ? ? 2 x3 2 x2 2 6x x2 1 2x 2 3 x 2 x 2 20 44. 45. x 1x 2 22 2 3 x 1x 1 12 2 2 ? 1 2 1 2 1 x x 1 7 2 3 x 2 1 x x 2 72 1 3 1x 1 12 x2 2 2x 2 3 3x 2 8 x2 1 6x 1 5 ? 4 2 2 21x 2 50x 2 16 x 2 3 7x 2 33x 2 10 46. 47. 48. 49. 50. 51. 52. a13 a 1 2 a 1 7a 1 12 a 2 16 a 2 56. 2 1 2 a 1a22 a 2 5a 1 4 x 1 5 7. 2 2 x 24 x12 58. 3 x12 1 x13 x13 3 x11 4x x21 6x x22 2 52x 3 x26 1 2 2 1 2 x12 x11 4 x21 3 x22 1 x25 2 62x 3 2 53. 1 x11 x21 2 b2 4 2 2 b 24 b 1 2b 2 3x 2 2 x 2 2 x 1 2x 1 1 x 21 2t t11 60. 2 2 2 t 2 25 t 1 5t 2 1 61. 2 131 y 21 y11 4 1 62. 2 1 2 2 t 24 t22 1 3 3x 2 2 63. 1 2 2 x22 x12 x 24 x 5 3 13x 2 12 64. 2 1 x23 x13 x2 2 9 1 1 x23 65. a 1 b? x22 x23 2x 1 1 1 2 66. a b4 x11 x22 x22 3x x 3x 1 1 67. 2 2 x24 x14 16 2 x2 7x 3x 3x 2 1 68. 1 1 2 x25 52x x 2 25 59. 69. x3 1 27 x2 1 4x 1 3 x2 1 x 2 6 4 4 a b x2 2 4 x2 1 2x x2 2 3x 1 9 3 x 1 x14 x24 70. 71. 2 1 2 1 2 2 1 2 x 1 3x 1 2 x 1 4x 1 3 x 1 5x 1 6 2 22 2 2z 2 2y 1 2 1y 2 x2 1z 2 x2 x2y x2z 3x 2 2 4x2 1 2 3x2 2 25 2 1 x2 1 x 2 20 x2 2 25 x2 2 16 72. 3x 1 2 x14 1 1 2 2 8x2 2 10x 2 3 6x 2 11x 1 3 4x 1 1 Simplify each complex fraction. Assume that no denominators are 0. 3a b 73. 6ac b2 3t2 9x 74. t 18x © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Appendix A72 Section 2 3a b ab 27 x2y ab 77. y2x ab © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 75. x2 2 5x 1 6 2x2y 78. x2 2 9 2x2y 1 1 1 x y 79. xy 1 1 1 x y 81. 1 1 2 x y 80. xy 11 11 1 x y 1 1 2 x y 82. 1 1 1 x y 2 3a 4a 2 b x 83. 1 1 1 b ax 12 6 x112 x 85. 6 x151 x 87. 89. 1 x1 x 86. x y x2 21 y2 2z 3 z 88. k5 1 1 1 1 k1 k2 Section 1 Section 2 98. Electronics The combined resistance R of three resistors with resistances of R1, R2, and R3 is given by the following formula. Simplify the complex fraction on the right side of the formula. R5 1 1 1 1 1 1 R1 R2 R3 Discovery and Writing 1 x 1 2 2x13 x 2x2 1 4 90. 4x 21 5 2x 1 1 x23 x22 92. 3 x 2 x23 x22 Simplify each complex fraction. Assume that no denominators are 0. x ab 99. 100. 1 3 1 1 21 2 1 21 3x 2a 1 y 101. 102. 1 2 11 21 1 2 11 21 x y a c ad 1 bc 1 5 is valid. b d bd a c a d 104. Explain why the formula 4 5 ? is valid. b d b c 105. Explain the Commutative Property of Addition and explain why it is useful. 106. Explain the Distributive Property and explain why it is useful. 103. Explain why the formula Review Write each expression without using negative exponents, and simplify the resulting complex fraction. Assume that no denominators are 0. 1 y21 93. 94. 1 1 x21 x21 1 y21 3 1x 1 22 21 1 2 1x 2 12 21 95. 1x 1 22 21 96. 97. Engineering The stiffness k of the shaft shown in the illustration is given by the following formula where k1 and k2 are the individual stiffnesses of each section. Simplify the complex fraction on the right side of the formula. x231 x 2 2 x12 x21 91. 3 x 1 x12 x21 84. 12 3xy 1 12 xy 3x Applications 3u2v 4t 76. 3uv .6 RationalExpressions A73 Write each expression without using absolute value symbols. 107. 0 26 0 108. 0 5 2 x 0 , given that x , 0 Simplify each expression. 109. a x3y22 23 b x21y Not For Sale 2x 1x 2 32 21 2 3 1x 1 22 21 1x 2 32 21 1x 1 22 21 111. "20 2 "45 112. 2 1x2 1 42 2 3 12x2 1 52 110. 127x62 2/3 A74 Appendix Not For Sale A Review of Basic Algebra APPENDIXpReview Sets of Real Numbers Definitions and Concepts Examples Natural numbers: The numbers that we count with. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, % Whole numbers: The natural numbers and 0. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, % Integers: The whole numbers and the negatives of the natural numbers. % , 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, % Rational numbers: {x 0 x can be written in the form ab 1b 2 02 , where a and b are integers.} 5 2 7 1 2 2 5 , 25 5 2 , , 2 , 0.25 5 1 1 3 3 4 All decimals that either terminate or repeat. 13 2 7 3 5 0.75, 5 2.6, 5 0.666 % 5 0.6, 5 0.63 4 5 3 11 Irrational numbers: Nonrational real numbers. All decimals that neither terminate nor repeat. "2, 2"26, p, 0.232232223 % Real numbers: Any number that can be expressed as a decimal. 26, 2 9 , 0, p, "31, 10.73 13 Prime numbers: A natural number greater than 1 that is divisible only by itself and 1. 2, 3, 5, 7, 11, 13, 17, % Composite numbers: A natural number greater than 1 that is not prime. 4, 6, 8, 9, 10, 12, 14, 15, % Even integers: The integers that are exactly divisible by 2. % , 26, 24, 22, 0, 2, 4, 6, % Odd integers: The integers that are not exactly divisible by 2. % , 25, 23, 21, 1, 3, 5, % Associative Properties: of addition 1a 1 b2 1 c 5 a 1 1b 1 c2 5 1 14 1 72 5 15 1 42 1 7 Commutative Properties: of addition a 1 b 5 b 1 a 4 1 7 5 7 1 4 1.7 1 2.5 5 2.5 1 1.7 of multiplication 1ab2 c 5 a 1bc2 of multiplication ab 5 ba 15 ? 42 ? 7 5 5 14 ? 72 4 ? 7 5 7 ? 4 1.7 12.52 5 2.5 11.72 Distributive Property: a 1b 1 c2 5 ab 1 ac 3 1x 1 62 5 3x 1 3 ? 6 0.2 1y 2 102 5 0.2y 2 0.2 1102 Open intervals have no endpoints. 123, 22 Double Negative Rule: 2 12a2 5 a Closed intervals have two endpoints. Half-open intervals have one endpoint. 2 1252 5 5 2 12a2 5 a 3 23, 2 4 123, 2 4 ( ) [ ] ( ] –3 –3 –3 2 2 2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. section A.1 Appendix Review A75 Definitions and Concepts Examples Absolute value: If x $ 0, then 0 x 0 5 x. 7 7 0 7 0 5 7 0 3.5 0 5 3.5 ` ` 5 0 0 0 5 0 2 2 Distance: The distance between points a and b on a number line is d 5 0b 2 a0. The distance on the number line between points with coordinates of 23 and 2 is d 5 0 2 2 1232 0 5 0 5 0 5 5. © Cengage Learning. All rights reserved. No distribution allowed without express authorization. If x , 0, then 0 x 0 5 2x. Exercises 7 7 0 27 0 5 7 0 23.5 0 5 3.5 ` 2 ` 5 2 2 Consider the set 5 26,23,0,12,3,p,"5,6,8 6 . List the numbers in this set that are 1. natural numbers. 16. 12x 1 y2 1 z 5 1y 1 2x2 1 z 2. whole numbers. 3. integers. 6. real numbers. Graph each subset of the real numbers: 18. the prime numbers between 10 and 20 19. the even integers from 6 to 14 Consider the set 5 26,23,0,12,3,p,"5,6,8 6 . List the numbers in this set that are 7. prime numbers. Graph each interval on the number line. 20. 23 , x # 5 8. composite numbers. 9. even integers. 10. odd integers. 2 1. x $ 0 or x , 21 4. rational numbers. 5. irrational numbers. Determine which property of real numbers justifies each statement. 11. 1a 1 b2 1 2 5 a 1 1b 1 22 12. a 1 7 5 7 1 a 13. 4 12x2 5 14 ? 22 x 14. 3 1a 1 b2 5 3a 1 3b 15. 15a2 7 5 7 15a2 17. 2 1262 5 6 2 2. 122, 4 4 2 3. 12`, 22 d 125,` 2 24. 12`,242 c 3 6,` 2 Write each expression without absolute value symbols. 26. 0 225 0 25. 0 6 0 2 7. 0 1 2 "2 0 28. 0 "3 2 1 0 29. On a number line, find the distance between points with coordinates of 25 and 7. Not For Sale Appendix A76 Not For Sale A Review of Basic Algebra Integer Exponents and Scientific Notation section A.2 Definitions and Concepts Examples Natural-number exponents: n factors of x x 5 x ? x ? x ? c? x x5 5 x ? x ? x ? x ? x Rules of exponents: If there are no divisions by 0, • 1xm2 n 5 xmn • xmxn 5 xm1n • 1xy2 n 5 xnyn • x0 5 1 1x 2 02 m • x 5 xm2n xn n 1 5 n x x • a b y 2n y 5a b x n Scientific notation: A number is written in scientific notation when it is written in the form N 3 10n, where 1 # 0 N 0 , 10. Exercises Write each expression without using exponents. 31. 125a2 2 30. 25a3 Write each expression using exponents. 32. 3ttt 33. 122b2 13b2 Simplify each expression. 35. 1p32 2 34. n2n4 36. 1x3y22 4 38. 1m n 2 23 0 2 a5 40. 8 a 4 1 1 2 5 6 36 60 5 1 622 5 x6 5 x624 5 x2 x4 x 23 6 3 63 216 a b 5a b 5 35 3 6 x x x Write each number in scientific notation. 386,000 5 3.86 3 105 0.0025 5 2.5 3 1023 Write each number in standard form. 7.3 3 103 5 7,300 5.25 3 1024 5 0.000525 3x2y22 22 b x2y2 23x3y 22 44. a b xy3 42. a a23b2 22 b ab23 2m22n0 23 45. a2 2 21 b 4m n 43. a 46. If x 5 23 and y 5 3, evaluate 2x2 2 xy2 Write each number in scientific notation. 47. 6,750 48. 0.00023 3 a b b2 p22q2 3 39. a b 2 a2 22 41. a 3 b b 37. a x 5 x5 a b 5 5 y y 1xy2 5 5 x5y5 n x x • a b 5 n y y • x2n 1x22 7 5 x2?7 5 x14 x2x5 5 x215 5 x7 Write each number in standard notation. 49. 4.8 3 102 50. 0.25 3 1023 51. Use scientific notation to simplify 145,0002 1350,0002 . 0.000105 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. n Appendix Review A77 Rational Exponents and Radicals section A.3 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Definitions and Concepts Examples Summary of a1/n definitions: • If a $ 0, then a1/n is the nonnegative number b such that bn 5 a. • If a , 0 and n is odd, then a1/n is the real number b such that bn 5 a. • If a , 0 and n is even, then a1/n is not a real number. Rule for rational exponents: If m and n are positive integers, a1/n is a real number, then m n n Properties of radicals: If all radicals are real numbers and there are no divisions by 0, then n n n • "ab 5 "a"b n a "a 5 n Åb "b m n mn n m Exercises 54. 132x52 1/5 56. 121,000x62 1/3 58. 1x12y22 1/2 60. a 53. a 2c2/3c5/3 1/3 b c22/3 Simplify each expression. 2/3 62. 64 64. a 66. a 3/4 16 b 81 8 22/3 b 27 68. 12216x32 2/3 1/3 27 b 125 55. 181a42 1/4 57. 1225x22 1/2 59. a 61. a x12 21/2 b y4 a21/4a3/4 21/2 b a9/2 23/5 63. 32 65. a 5 5 5 10 " 32x10 5 " 32" x 5 2x2 4 12 x12 " x x12/4 x3 5 4 5 5 Å 625 5 5 "625 4 70. "36 72. 9 Å 25 73. 74. "x2y4 76. m8n4 Å p16 4 77. Simplify and combine terms. 78. "50 1 "8 3 3 4 3 27 Å 125 3 3 75. " x 4 a15b10 Å c5 5 79. "12 1 "3 2 "27 Rationalize each denominator. 2/5 32 b 243 81. 83. "7 "5 1 3 " 2 Not For Sale 6 "64 5 " 64 5 2 71. 2"49 80. "24x 2 "3x 16 23/4 b 625 pa/2pa/3 69. pa/6 67. a 2#3 Simplify each expression. Simplify each expression, if possible. 3 " 125 5 1251/3 5 5 3 3 3 # "64 5 " 8 5 2 #" 64 5 "4 5 2 • #"a 5 #"a 5"a 52. 1211/2 12162 1/2 is not a real number because no real number squared is 216. 82/3 5 181/32 2 5 22 5 4 or 1822 1/3 5 641/3 5 4 Definition of "a: n "a 5 a1/n n 12272 1/3 5 23 because 1232 3 5 227 is in lowest terms, and am/n 5 1a1/n2 m 5 1am2 1/n • 161/2 5 4 because 42 5 16 82. 84. 8 "8 2 3 " 25 Appendix A78 Not For Sale A Review of Basic Algebra Rationalize each numerator. "2 5 section A.4 86. "5 5 87. "2x 3 88. 3 3" 7x 2 Polynomials Definitions and Concepts Examples Monomial: A polynomial with one term. 2, 3x, 4x2y, 2x3y2z Binomial: A polynomial with two terms. 2t 1 3, 3r2 2 6r, 4m 1 5n Trinomial: A polynomial with three terms. 3p2 2 7p 1 8, 3m2 1 2n 2 p The degree of a monomial is the sum of the exponents on its variables. The degree of 4x2y3 is 2 1 3 5 5 The degree of a polynomial is the degree of the term in the polynomial with highest degree. The degree of the first term of 3p2q3 2 6p3q4 1 9pq2 is 2 1 3 5 5, the degree of the second term is 3 1 4 5 7, and the degree of the third term is 1 1 2 5 3. The degree of the polynomial is the largest of these. It is 7. Multiplying a monomial times a polynomial: a 1b 1 c 1 d 1 c 2 5 ab 1 ac 1 ad 1 c Addition and subtraction of polynomials: To add or subtract polynomials, remove parentheses and combine like terms. 3 1x 1 22 5 3x 1 6 2x 13x2 2 2y 1 32 5 6x3 2 4xy 1 6x Add: 13x2 1 5x2 1 12x2 2 2x2 . 13x2 1 5x2 1 12x2 2 2x2 5 3x2 1 5x 1 2x2 2 2x 5 3x2 1 2x2 1 5x 2 2x 5 5x2 1 3x Subtract: 14a2 2 5b2 2 13a2 2 7b2 . 14a2 2 5b2 2 13a2 2 7b2 5 4a2 2 5b 2 3a2 1 7b 5 4a2 2 3a2 2 5b 1 7b 5 a2 1 2b Special products: 1x 1 y2 2 5 x2 1 2xy 1 y2 1x 2 y2 2 5 x2 2 2xy 1 y2 1x 1 y2 1x 2 y2 5 x2 2 y2 Multiplying a binomial times a binomial: Use the FOIL method. 12m 1 32 2 5 4m2 1 12m 1 9 14t 2 3s2 2 5 16t2 2 24ts 1 9s2 12m 1 n2 12m 2 n2 5 4m2 2 2mn 1 2mn 2 n2 5 4m2 2 n2 12a 1 b2 1a 2 b2 5 2a 1a2 1 2a 12b2 1 ba 1 b 12b2 5 2a2 2 2ab 1 ab 2 b2 5 2a2 2 ab 2 b2 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 85. Appendix Review A79 Definitions and Concepts Examples The conjugate of a 1 b is a 2 b. To rationalize the denominator of a radical expression, multiply both the numerator and denominator of the rational expression by the conjugate of the denominator. Rationalize the denominator: x © Cengage Learning. All rights reserved. No distribution allowed without express authorization. "x 1 2 5 x 1"x 2 22 x "x 1 2 1"x 1 22 1"x 2 22 x"x 2 2x 5 x24 Division of polynomials: To divide polynomials, use long division. Give the degree of each polynomial and tell whether the polynomial is a monomial, a binomial, or a trinomial. 89. x3 2 8 90. 8x 2 8x2 2 8 91. "3x2 92. 4x4 2 12x2 1 1 Perform the operations and simplify. 93. 2 1x 1 32 1 3 1x 2 42 94. 3x2 1x 2 12 2 2x 1x 1 32 2 x2 1x 1 22 95. 13x 1 22 13x 1 22 96. 13x 1 y2 12x 2 3y2 97. 14a 1 2b2 12a 2 3b2 98. 1z 1 32 13z2 1 z 2 12 99. 1an 1 22 1an 2 12 2 100. Q"2 1 xR 101. Q"2 1 1RQ"3 1 1R 3 3 3 3 2 2RQ" 9 1 2" 3 1 4R 102. Q" Multiply the numerator and denominator by the conjugate of "x 1 2. Simplify. Divide: 2x 1 3q6x3 1 7x2 2 x 1 3. 3x2 2x 1 3q 6x3 1 7x2 6x3 1 9x2 2 2x2 2 2x2 Exercises . 2 x11 2 x13 2 x 2 3x 1 2x 1 3 1 2x 1 3 0 Rationalize each denominator. 2 22 104. 103. "3 2 1 "3 2 "2 105. 2x "x 2 "y 106. Rationalize each numerator. "x 1 2 107. 5 108. 1 2 "a a 110. 4a2b3 1 6ab4 2b2 "x 2 2 Perform each division. 3x2y2 109. 3 6x y "x 1 "y 111. 2x 1 3q2x3 1 7x2 1 8x 1 3 112. x2 2 1qx5 1 x3 2 2x 2 3x2 2 3 Not For Sale Appendix A80 Not For Sale A Review of Basic Algebra Factoring Polynomials section A.5 Definitions and Concepts Examples ab 1 ac 5 a 1b 1 c2 3p3 2 6p2q 1 9p 5 3p 1p2 2 2pq 1 32 x2 2 y2 5 1x 1 y2 1x 2 y2 4x2 2 9 5 12x 1 32 12x 2 32 Factoring the difference of two squares: Factoring trinomials: • Trinomial squares x2 1 2xy 1 y2 5 1x 1 y2 2 9a2 1 12ab 1 4b2 5 13a 1 2b2 13a 1 2b2 5 13a 1 2b2 2 x2 2 2xy 1 y2 5 1x 2 y2 2 • To factor general trinomials use trial and error or grouping. Factoring the sum and difference of two cubes: x3 1 y3 5 1x 1 y2 1x2 2 xy 1 y22 27a3 2 8b3 5 13a 2 2b2 19a2 1 6ab 1 4b22 Exercises Factor each expression completely, if possible. 2 6x2 2 5x 2 6 5 12x 2 32 13x 1 22 r3 1 8 5 1r 1 22 1r2 2 2r 1 42 x3 2 y3 5 1x 2 y2 1x2 1 xy 1 y22 113. 3t3 2 3t r2 2 4rs 1 4s2 5 1r 2 2s2 1r 2 2s2 5 1r 2 2s2 2 114. 5r3 2 5 119. x2 1 6x 1 9 2 t2 120. 3x2 2 1 1 5x 121. 8z3 1 343 122. 1 1 14b 1 49b2 123. 121z2 1 4 2 44z 124. 64y3 2 1,000 2 115. 6x 1 7x 2 24 116. 3a 1 ax 2 3a 2 x 117. 8x3 2 125 118. 6x2 2 20x 2 16 125. 2xy 2 4zx 2 wy 1 2zw 126. x8 1 x4 1 1 D section A.6 efinitions and Concepts Properties of fractions: If there are no divisions by 0, then a c • 5 if and only if ad 5 bc. b d a ax • 5 b bx a c ac • ? 5 b d bd a c ad • 4 5 b d bc Algebraic Fractions xamples 3x 6x 5 because 13x2 8 and 4 16x2 both equal 24x. 4 8 6x2 3?2?x?x 3 5 3 5 8x 2?4?x?x?x 4x 3p 2p 3p ? 2p 3p ? 2p p2 ? 5 5 5 2 2q 6q 2q ? 6q 2q ? 3 ? 2q 2q 2t 2t 2t 6s 2t ? 6s 2t ? 2 ? 3s 4 5 ? 5 5 52 3s 6s 3s 2t 3s ? 2t 3s2t © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Factoring out a common monomial: E Appendix Review A81 Definitions and Concepts a c a1c a c a2c 1 5 • 2 5 b b b b b b a a • a ? 1 5 a • 5 a • 5 1 1 a a 2a a 2a • 5 52 52 b 2b 2b b a a 2a 2a 5 52 • 2 5 b 2b b 2b © Cengage Learning. All rights reserved. No distribution allowed without express authorization. • Simplifying rational expressions: To simplify a rational expression, factor the numerator and denominator, if possible, and divide out factors that are common to the numerator and denominator. Adding or subtracting rational expressions: To add or subtract rational expressions with unlike denominators, find the LCD of the expressions, write each expression with a denominator that is the LCD, add or subtract the expressions, and simplify the result, if possible. Examples x y x1y 3p p 3p 2 p 2p p 1 5 2 5 5 5 4 4 4 2q 2q 2q 2q q 7 7 7 ? 1 5 7 5 7 51 1 7 7 27 7 27 5 52 52 2 22 22 2 7 7 27 27 2 5 5 52 2 22 2 22 2 2 x To simplify 2x 2 4 , factor 21 from the numerator and 2 from the denominator to get 21 122 1 x2 21 1x 2 22 21 1 22x 5 5 5 52 2x 2 4 2 1x 2 22 2 1x 2 22 2 2 2x 2x 2x 1x 2 32 2x 1x 1 22 2 5 2 1x 1 22 1x 2 32 1x 2 32 1x 1 22 x12 x23 2 1 1 2x x 2 3 2 2x x 1 22 5 1x 1 22 1x 2 32 2x2 2 6x 2 2x2 2 4x 1x 1 22 1x 2 32 210x 5 1x 1 22 1x 2 32 5 Complex fractions: To simplify complex fractions, multiply the numerator and denominator of the complex fraction by the LCD of all the fractions. y y 2ya1 2 b 2 2 2y 2 y2 y 12 2 y2 5 5 5 1 1 1 1 21y 21y 1 2ya 1 b y 2 y 2 12 Exercises Simplify each rational expression. 22x a2 2 9 127. 2 128. 2 x 2 4x 1 4 a 2 6a 1 9 Perform each operation and simplify. Assume that no denominators are 0. 133. x2 1 x 2 6 x2 2 x 2 6 x2 2 4 ? 2 4 2 2 x 2x26 x 1x22 x 2 5x 1 6 2x 1 6 2x2 2 2x 2 4 x2 2 x 2 2 4 b 2 x15 x2 2 25 x 2 2x 2 15 2 3x 135. 1 x24 x15 134. a 129. x2 2 4x 1 4 x2 1 5x 1 6 ? x12 x22 130. 2y2 2 11y 1 15 y2 2 2y 2 8 ? 2 y2 2 6y 1 8 y 2y26 136. 5x 3x 1 7 2x 1 1 2 1 x22 x12 x12 131. 2t2 1 t 2 3 10t 1 15 4 2 2 3t 2 7t 1 4 3t 2 t 2 4 137. x x x 1 1 x21 x22 x23 132. p2 1 7p 1 12 p2 2 9 4 p3 1 8p2 1 4p p2 138. Not For Sale x 3x 1 7 2x 1 1 2 1 x11 x12 x12 139. 140. Not For Sale Appendix A Review of Basic Algebra 3 1x 1 12 5 1x2 1 32 x 2 1 2 x x x11 3x x2 1 4x 1 3 x2 1 x 2 6 1 2 2 x11 x 1 3x 1 2 x2 2 4 Simplify each complex fraction. Assume that no denominators are 0. 5x 2 141. 2 3x 8 3x y 142. 6x y2 1 1 1 x y 143. x2y 144. x21 1 y21 y21 2 x21 APPENDIXpTest Consider the set 5 27,223,0,1,3,"10,4 6 . 1. List the numbers in the set that are odd integers. 2. List the numbers in the set that are prime numbers. Determine which property justifies each statement. 3. 1a 1 b2 1 c 5 1b 1 a2 1 c 4. a 1b 1 c2 5 ab 1 ac Graph each interval on a number line. 6. 12`, 232 c 3 6,` 2 5. 24 , x # 2 Write each expression without using absolute value symbols. 8. 0 x 2 7 0 , when x , 0 7. 0 217 0 Find the distance on a number line between points with the following coordinates. 9. 24 and 12 10. 220 and 212 Simplify each expression. Assume that all variables represent positive numbers, and write all answers without using negative exponents. r2r3s 11. x4x5x2 12. 4 2 rs 1a21a22 22 x0x2 6 13. 14. a 22 b 23 a x Write each number in scientific notation. 15. 450,000 16. 0.000345 Write each number in standard notation. 17. 3.7 3 103 18. 1.2 3 1023 Simplify each expression. Assume that all variables represent positive numbers, and write all answers without using negative exponents. 36 3/2 20. a b 19. 125a42 1/2 81 8t6 22/3 3 21. a 9 b 22. "27a6 27s 2 3. "12 1 "27 3 3 24. 2" 3x4 2 3x" 24x 25. Rationalize the denominator: 26. Rationalize the numerator: Perform each operation. 27. 1a2 1 32 2 12a2 2 42 28. 13a3b22 122a3b42 29. 13x 2 42 12x 1 72 30. 1an 1 22 1an 2 32 31. 1x2 1 42 1x2 2 42 32. 1x2 2 x 1 22 12x 2 32 33. x 2 3q6x2 1 x 2 23 34. 2x 2 1q2x3 1 3x2 2 1 Factor each polynomial. 35. 3x 1 6y 36. x2 2 100 37. 10t2 2 19tw 1 6w2 x "x 2 2 "x 2 "y "x 1 "y . . © Cengage Learning. All rights reserved. No distribution allowed without express authorization. A82 AppendixTest A83 38. 3a3 2 648 39. x4 2 x2 2 12 40. 6x4 1 11x2 2 10 Perform each operation and simplify if possible. Assume that no denominators are 0. x 2 41. 1 x12 x12 © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 42. x x 2 x11 x21 43. x2 1 x 2 20 x2 2 25 ? x2 2 16 x25 44. x12 x2 2 4 4 x 1 2x 1 1 x11 2 Simplify each complex fraction. Assume that no denominators are 0. 1 1 1 a b x21 45. 46. 21 1 x 1 y21 b Not For Sale © Cengage Learning. All rights reserved. No distribution allowed without express authorization. Not For Sale