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LESSON 6.4 Name Factoring Polynomials Class 6.4 Date Factoring Polynomials Essential Question: What are some ways to factor a polynomial, and how is factoring useful? Common Core Math Standards The student is expected to: Resource Locker A-SSE.2 Explore Use the structure of an expression to identify ways to rewrite it. Also N-CN.8(+), A-SSE.1a, A-APR.3, A-CED.1 Factoring a polynomial of degree n involves finding factors of a lesser degree that can be multiplied together to produce the polynomial. When a polynomial has degree 3, for example, you can think of it as a rectangular prism whose dimensions you need to determine. Mathematical Practices MP.8 Patterns A Language Objective R Yellow(Y): V = 8x Total volume: V = x 3 + 6x 2 + 8x x+ 2 B The volume of the red piece is found by cubing the length of one edge. What is the height of this piece? © Houghton Mifflin Harcourt Publishing Company x C The volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height of the prism. Notice that the green prism shares two dimensions with the cube. What are these dimensions? x and x D What is the length of the third edge of the green prism? 2x Divide by both of the known dimensions to find the missing edge length. ____ x· x =2 2 The length of the last edge is 2. E x and the width of the green prism You showed that the width of the cube is 2 is . What is the width of the entire prism? x+2 Module 6 be ges must EDIT--Chan DO NOT Key=NL-C;CA-C Correction Lesson 4 309 gh “File info” made throu Date Class Name Factoring 6.4 ion: What Quest Essential are some s Polynomial ways to factor , and how a polynomial y ways to sion to identif re of an expres the structu .1 A-SSE.2 Use A-APR.3, A-CED A-SSE.1a, rewrite it. is factoring useful? of can think finding factors for example, you 3, n involves of degree has degree a polynomial When a polynomial Factoring polynomial. determine. produce the sions you need to as follows: prism are whose dimen parts es of the The volum 3 =x Red(R): V 2 V = 2x Green(G): Yellow(Y): y g Compan Publishin Harcour t Volume 2 8x + 6x + piece is found by cubing the length 2 = x3 + 6x What of one edge. x+ 4 + 8x is the height of x width, and , w is the the the length sions with where l is two dimen is V = lwh, prism shares gular prism the green e of a rectan . Notice that The volum of the prism h is the height these dimensions? are cube. What x and x © Houghto G x x+ 2 3 e of the red The volum this piece? n Mifflin R 2 = 4x e: V = x Total volum Turn to these pages to find this lesson in the hardcover student edition. Y B gular of the rectan V = 8x Blue(B): V HARDCOVER PAGES 223230 Resource Locker (+), Also N-CN.8 Model a Visual n together to Analyzing ial Factorizatio multiplied that can be as a rectangular prism of it for Polynom a lesser degree Explore ? h. x · x = green prism edge lengt edge of the missing the third find the length of nsions to What is the known dime both of the Divide by is 2. last edge prism h of the of the green The lengt x and the width is cube the width of prism? d that the You showe of the entire 2 . What is the width is x+2 2 2x ____ 2 Lesson 4 309 Module 6 L4 309 4_U3M06 SE38589 A2_MNLE Lesson 6.4 x+ 4 Volume = x3 + 6x2 + 8x A2_MNLESE385894_U3M06L4 309 309 G x Blue(B): V = 4x 2 Essential Question: What are some ways to factor a polynomial, and how is factoring useful? View the Engage section online. Discuss the photo and how the volume of the flower bed is related to its outer and inner dimensions. Then preview the Lesson Performance Task. Y B Green(G): V = 2x 2 ENGAGE PREVIEW: LESSON PERFORMANCE TASK The volumes of the parts of the rectangular prism are as follows: Red(R): V = x 3 Work with a partner to complete a chart detailing how factoring can be used. Possible answer: Methods include factoring out the greatest common monomial factor, recognizing special factoring patterns, and factoring by grouping. Factoring is useful when solving polynomial equations because the zero-product property can be applied to the factored polynomial on one side of the equation as long as the other side is 0. Analyzing a Visual Model for Polynomial Factorization 6/8/15 11:43 PM 6/8/15 11:41 PM You determined that the length of the green piece is x. Use the volume of the yellow piece and the information you have derived to find the length of the prism. EXPLORE Since the volume of the yellow piece is 8x, the height of the yellow piece is x, and the width of the yellow piece is 2, simply divide out to find that the last remaining edge length is 4. Analyzing a Visual Model for Polynomial Factorization INTEGRATE TECHNOLOGY x Students have the option of completing the polynomial factorization activity either in the book or online. x+ 4 x+ 2 Volume = x3 + 6x2 + 8x Since the dimensions of the overall prism are x, x + 2, and x + 4, the volume of the overall prism can be rewritten in factored form as V = (x)(x + 2)(x + 4). Multiply these polynomials together to verify that this is equal to the original given expression for the volume of the overall figure. QUESTIONING STRATEGIES How is factoring useful when determining the variable dimensions of a polynomial model like a rectangular prism? Factoring is useful when the volumes of parts of a three-dimensional model are known and can be added to get the variable dimensions of the figure. Since factoring is the opposite of multiplying, the polynomial can then be written in factored form and multiplied to give the volume of the whole figure. V = x 3 + 6x 2 + 8x Reflect 1. Discussion What is one way to double the volume of the prism? Sample Answer: One way to double the volume of the prism is to double the height of the prism. Explain 1 Factoring Out the Greatest Common Monomial First Factor each polynomial over the integers. Example 1 6x 3 + 15x 2 + 6x 6x + 15x + 6x Write out the polynomial. x(6x + 15x + 6) Factor out a common monomial, an x. 3 2 2 3x(2x + 5x + 2) 2 3x(2x + 1)(x + 2) © Houghton Mifflin Harcourt Publishing Company Most polynomials cannot be factored over the integers, which means to find factors that use only integer coefficients. But when a polynomial can be factored, each factor has a degree less than the polynomial’s degree. While the goal is to write the polynomial as a product of linear factors, this is not always possible. When a factor of degree 2 or greater cannot be factored further, it is called an irreducible factor. EXPLAIN 1 Factoring Out the Greatest Common Monomial First Factor out a common monomial, a 3. AVOID COMMON ERRORS Factor into simplest terms. Students often forget to factor out the greatest common monomial as the first step in factoring. Point out that if the terms of a polynomial have a greatest common factor, it almost always should be factored first, before analyzing the remainder of the terms for factoring. Emphasize that sometimes finding the greatest common factor is the only way to factor some polynomials. Note: The second and third steps can be combined into one step by factoring out the greatest common monomial. Module 6 310 Lesson 4 PROFESSIONAL DEVELOPMENT A2_MNLESE385894_U3M06L4 310 Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.8, which calls for students to “look for and express regularity in repeated reasoning.” Students are already familiar with multiplying polynomials but, in this lesson, they must analyze the conditions that help them factor a polynomial, or rewrite it as the product of individual factors of lesser degree. These factors, when multiplied, give the original polynomial. Many methods of factoring, including special factoring patterns, are presented so that students can analyze the polynomial and explain which method gives the factorization more easily. Factoring polynomials is a useful tool for solving polynomial equations by using the zero-product property. 6/27/14 2:37 PM Factoring Polynomials 310 QUESTIONING STRATEGIES How can you tell if the terms of a polynomial have a greatest common factor? Each of the terms has a number that divides all of the terms and/or a power of a variable that divides each term. The common monomial factor is a product of this number and the power of one or more of the variables. 2x 3 - 20x 2x 3 - 20 x Write out the polynomial. 2x (x 2 - 10) Factor out the greatest common monomial. Reflect 2. Why wasn’t the factor x 2 - 10 further factored? Since 10 does not have any two factors that sum to 0, (x 2 - 10) is irreducible over the integers. 3. What does it mean for a polynomial to be “factored completely”? The greatest monomial factor is factored out and the remaining polynomial is factored into factors that cannot be factored further: they are irreducible. Consider what happens when you factor x 2 - 10 over the real numbers and not merely the integers. Find a such that x 2 - 10 = (x - a)(x + a). a = √10 ― Your Turn 4. 3x 3 + 7x 2 + 4x x(3x 2 + 7x + 4) Factor out a common monomial, an x. x(3x + 4)(x + 1) EXPLAIN 2 Explain 2 Factor into simplest terms. Recognizing Special Factoring Patterns Remember the factoring patterns you already know: Recognizing Special Factoring Patterns Difference of two squares: a 2 - b 2 = (a + b)(a - b) Perfect square trinomials: a 2 + 2ab + b 2 = (a + b) and a 2 - 2ab + b 2 = (a - b) 2 2 There are two other factoring patterns that will prove useful: they learned in Algebra 1, including: difference of two squares: a 2 - b 2 = (a + b)(a - b); and perfect 2 square trinomials: a 2 + 2ab + b 2 = (a + b) and 2 a 2 - 2ab + b 2 = (a - b) . Emphasize that there are two more useful factoring patterns in this lesson: sum of two cubes: a 3 + b 3 = (a + b)(a 2 - ab + b 2) and difference of two cubes: a 3 - b 3 = (a - b)(a 2 + ab + b 2). Students should see that each of the factors is irreducible. Sum of two cubes : a 3 + b 3 = (a + b)(a 2 - ab + b 2) © Houghton Mifflin Harcourt Publishing Company INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students should review the factoring patterns Difference of two cubes : a 3 - b 3 = (a - b)(a 2 + ab + b 2) Notice that in each of the new factoring patterns, the quadratic factor is irreducible over the integers. Example 2 Factor the polynomial using a factoring pattern. 27x 3 + 64 27x 3 + 64 Write out the polynomial. 27x 3 = (3x) 64 = (4) 3 Check if 27x 3 is a perfect cube. 3 Check if 64 is a perfect cube. a 3 + b 3 = (a + b)(a 2 - ab + b 2) Use the sum of two cubes formula to factor. (3x) + 4 3 = (3x +4)((3x) - (3x)(4) + 4 2) 3 2 27x 3 + 64 = (3x + 4)(9x 2 - 12x + 16) Module 6 311 Lesson 4 COLLABORATIVE LEARNING A2_MNLESE385894_U3M06L4.indd 311 Small Group Activity Help groups of students review the factoring patterns they learned in previous courses along with how to factor the sum or difference of two cubes. Provide each student with an example of one type of factoring, and ask them to show and explain the first step in factoring their problem. Then they pass the problem to another student, who writes the next step and explains it. They continue to pass the problem on until each problem is completely factored and all steps are explained. Encourage them to use these as examples of the types of factoring when they write about factoring in their journals. 311 Lesson 6.4 3/19/14 1:37 PM B 8x 3 - 27 3 8 x - 27 QUESTIONING STRATEGIES Write out the polynomial. 8x 3 = ( 2 x) 3 Check if 8x 3 is a perfect cube. 27 = ( 3 ) 3 Check if 27 is a perfect cube. a 3 - b 3 = (a - b)(a 2 + ab + b 2) What are the steps for using the sum or difference of cubes formulas to factor? Factor out the greatest monomial factor, rewrite as the sum or difference of perfect cubes, and then apply the sum or difference of cubes formula. Use the difference of two cubes formula to factor. 8x 3 - 27 = ( 2 x - 3 )( 4 x 2 + 6 x + 9 ) Reflect 5. The equation 8x 3 - 27 = 0 has three roots. How many of them are real, what are they, and how many are nonreal? To find the roots of the given expression, notice that the expression is the difference of cubes. Use the difference of cubes formula to factor 8x 3 - 27 8x 3 - 27 = (2x-3)(4x 2 + 6x + 9) To find the first root, set 2x - 3 equal to 0 and solve for x. 2x - 3 = 0 2x = 3 3 x = __ 2 3 is a real root. Next, look at 4x 2 + 6x + 9 Notice that because the polynomial is Therefore __ 2 irreducible over the integers, the quadratic formula must be used to solve for the roots. ――――― -6± √6 2 - 4(4)(9) x = ________________ 2(4) ――― -6± √36 - 144 = _____________ 8 ―― -6± √-108 = ___________ 8 Your Turn 6. 40x 4 + 5x 40x 4 + 5x Write out the polynomial. 5x(8x + 1) 3 Factor out 5x. 8x = (2x) 3 1 = (1) 3 Check if 8x 3 is a perfect cube. 3 Check if 1 is a perfect cube. a + b = (a + b)(a - ab + b 3 3 2 2 ) © Houghton Mifflin Harcourt Publishing Company As the number under the square root symbol is negative, both of the values of x will be 3 and two roots are nonreal. nonreal numbers. Therefore, the one real root is __ 2 Use the formula to factor. 8x + 1 = (2x + 1)(4x - 2x + 1) 3 2 5x(8x 3 + 1) = 5x(2x + 1)(4x 2 - 2x + 1) 40x 4 + 5x = 5x(2x + 1)(4x 2 - 2x + 1) Module 6 312 Lesson 4 DIFFERENTIATE INSTRUCTION A2_MNLESE385894_U3M06L4 312 Communicating Math 6/27/14 2:40 PM Give groups of students several different polynomials to factor. Have all students explain their reasoning or describe the procedures they used to factor the polynomial completely. Encourage them to make visual aids or graphic organizers to help them with their explanations. Ask for a volunteer from each group to summarize to the class how to factor a polynomial. Factoring Polynomials 312 Explain 3 EXPLAIN 3 Another technique for factoring a polynomial is grouping. If the polynomial has pairs of terms with common factors, factor by grouping terms with common factors and then factoring out the common factor from each group. Then look for a common factor of the groups in order to complete the factorization of the polynomial. Factoring by Grouping Example 3 AVOID COMMON ERRORS Students may not have a systematic approach to factoring. They will find it easier to factor by grouping if they use a strategy of looking for a common monomial factor, checking the factoring patterns, checking if factoring by grouping applies to their problem, and so on. QUESTIONING STRATEGIES product property in the context of factoring and solving a real-world polynomial equation. Tell them that once a polynomial equation is established that models the real-world situation, then they apply the known information to the polynomial and rewrite it in a form (with zero on one side) that makes it possible to factor and solve the polynomial using the zero-product property. 313 Lesson 6.4 Write out the polynomial. x3 - x 2 + x - 1 Group by common factor. (x3 - x 2) + (x - 1) Factor. x 2(x - 1) + 1(x - 1) Regroup. (x 2 + 1)(x - 1) x4 + x3 + x + 1 Write out the polynomial. x4 + x3 + x + 1 Group by common factor. 4 3 ( x + x ) + (x + 1) x 3 (x + 1) + 1 (x + 1) 3 ( x + 1 )(x + 1) 2 ( x - x + 1)(x + 1)(x + 1) Apply sum of two cubes to the first term. 2 ( x + 1 ) 2( x - x + 1) Substitute this into the expression and simplify. Your Turn x 3 + 3x 2 + 3x + 2 x 3 + 3x 2 + 3x + 2 © Houghton Mifflin Harcourt Publishing Company INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Discuss with students how to use the zero- x3 + x 2 + x + 1 Regroup. 7. Solving a Real-World Problem by Factoring a Polynomial Factor the polynomial by grouping. Factor. How do you factor a polynomial by grouping? Rearrange the terms so that when they are grouped they will have common factors; group the terms; factor each group, using factoring patterns if necessary; then rearrange and assemble the factors using the distributive property. EXPLAIN 4 Factoring by Grouping (x 3 + 2x ) + (x 2 3 +3x + 3) x (x +2) + (x + 1)(x + 2) 2 (x 2 + x + 1)(x + 2) Explain 4 Write out the polynomial. Group by common factor. Factor. Regroup. Solving a Real-World Problem by Factoring a Polynomial Remember that the zero-product property is used in solving factorable quadratic equations. It can also be used in solving factorable polynomial equations. Module 6 A2_MNLESE385894_U3M06L4 313 313 Lesson 4 6/27/14 2:41 PM Example 4 Write and solve a polynomial equation for the situation described. QUESTIONING STRATEGIES A water park is designing a new pool in the shape of a rectangular prism. The sides and bottom of the pool are made of material 5 feet thick. The length must be twice the height (depth), and the interior width must be three times the interior height. The volume of the box must be 6000 cubic feet. What are the exterior dimensions of the pool? How is the zero-product property used to solve a polynomial equation? The polynomial equation is rewritten so that 0 is on one side, and then the equation is factored. The zero-product property states that if each factor is set equal to zero and the resulting equation solved, then the values obtained are solutions to the original polynomial equation. The dimensions of the interior of the pool, as described by the problem, are the following: h=x-5 w = 3x - 15 l = 2x - 10 The formula for volume of a rectangular prism is V = lwh. Plug the values into the volume equation. V = (x - 5)(3x - 15)(2x - 10) V = (x - 5)(6x 2 - 60x + 150) V = 6x 3 - 90x 2 + 450x - 750 Now solve for V = 6000. 6000 = 6x 3 - 90x 2 + 450x - 750 0 = 6x 3 - 90x 2 + 450x - 6750 Factor the resulting new polynomial. 6x 3 - 90x 2 + 450x - 6750 = 6x 2(x - 15) + 450(x - 15) = (6x 2 + 450)(x - 15) The only real root is x = 15. Engineering To build a hefty wooden feeding trough for a zoo, its sides and bottom should be 2 feet thick, and its outer length should be twice its outer width and height. What should the outer dimensions of the trough be if it is to hold 288 cubic feet of water? Volume = Interior Length(feet) ⋅ Interior Width(feet) ⋅ Interior Height(feet) 288 = ( 2x - 4)( x - 4)( x - 2) 288 = 2 x 3 - 16 x 2 + 40 x - 32 0 = 2 x 3 - 16 x 2 + 40 x - 320 2 0 = 2x (x - 8 ) + 40 (x - 8 ) 0= 2 (x 2 + 20 )(x - 8 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©morrison77/Shutterstock The interior height of the pool will be 10 feet, the interior width 30 feet, and the interior length 20 feet. Therefore, the exterior height is 15 feet, the exterior length is 30 feet, and the exterior width is 40 feet. ) The only real solution is x = 8 . The trough is 16 feet long, 8 feet wide, and 8 feet high. Module 6 A2_MNLESE385894_U3M06L4.indd 314 314 Lesson 4 3/19/14 1:37 PM Factoring Polynomials 314 Your Turn ELABORATE 8. QUESTIONING STRATEGIES Volume = Interior Length(feet) ⋅ Interior Width(feet) ⋅ Interior Height(feet) How do you determine whether a polynomial is not factorable? First determine that the terms do not have a common factor, that no factoring pattern applies to the terms of the polynomial, and that there is no other way to rewrite the polynomial as the product of irreducible factors. 972 = (2x − 6)(x − 6)(x − 3) 972 = 2x 3 - 24x 2 + 90x - 108 0 = 2x 3 - 24x 2 + 90x - 1080 0 = 2x 2(x - 12) + 90(x - 12) 0 = 2(x 2 + 45)(x - 12) The only real solution is x = 12. The shed is 24 feet long, 12 feet wide, and 12 feet high. Elaborate 9. COGNITIVE STRATEGIES Describe how the method of grouping incorporates the method of factoring out the greatest common monomial. After grouping an expression into separate parts, you factor out the greatest common monomial of each part to reveal a polynomial factor that is common to all the To help students remember the formulas for the sum and difference of cubes, you may wish to use the mnemonic device SOPPS for the order of the terms in the second factor: separate parts. 10. How do you decide if an equation fits in the sum of two cubes pattern? If there are two terms in the equation being added and these terms are perfect cubes, then it fits the sum of two cubes pattern. Square Opposite-sign Product Plus Square. 11. How can factoring be used to solve a polynomial equation of the form p(x) = a, where a is a nonzero constant? Subtract a to get p(x) - a = 0. Factor out common monomials or use grouping to factor SUMMARIZE THE LESSON the polynomial. © Houghton Mifflin Harcourt Publishing Company Have students make a graphic organizer that lists their own methods of factoring a polynomial completely. The lists should include factoring out the greatest monomial factor, applying rules for factoring the sum and difference of two cubes, and previously learned rules such as factoring the difference of two squares. Engineering A new shed is being designed in the shape of a rectangular prism. The shed’s side and bottom should be 3 feet thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the shed be if it is to have 972 cubic feet of space? 12. Essential Question Check-In What are two ways to factor a polynomial? Recognizing special factoring patterns and factoring by grouping Module 6 Lesson 4 315 LANGUAGE SUPPORT A2_MNLESE385894_U3M06L4 315 16/10/14 9:31 AM Connect Vocabulary Have students work together to complete a table like the one shown. Useful Ways to Factor Polynomials Perfect square trinomials: Sum of two cubes: Difference of two cubes: 315 Lesson 6.4 a 2 + 2ab + b 2 = (a + b) and 2 a 2 - 2ab + b 2 = (a - b) 2 a 3 + b 3 = (a + b)(a 2 - ab + b 2) a 3 - b 3 = (a - b)(a 2 + ab + b 2) EVALUATE Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice Factor the polynomial, or identify it as irreducible. 1. x 3 + x 2 - 12x 2. x(x + x - 12) 2 x3 + 5 Irreducible. x(x + 4)(x - 3) 3. x 3 - 125 4. x = (1x) 3 125 = (5) ( 8x 3 + 125 125 = (5) 216x 3 + 64 8. 8(27x 3 + 8) 27x = (3x) 3 3 8 = (2) 2x 3 + 6x Example 1 Factoring Out the Greatest Common Monomial Factor First Exercises 1–2, 4, 6, 10–12 8x 3 - 64 Example 2 Recognizing Special Factoring Patterns Exercises 3, 5, 7–9 8(x - 2)(x 2 - 2x + 4) Example 3 Factoring by Grouping Exercises 13–18 Example 4 Solving a Real-World Problem by Factoring a Polynomial Exercises 19–22 2x(x 2 + 3) 8x 3 + 125 = (2x + 5)(4x 2 - 10x + 25) 8(x 3 - 8) 3 a 3 + b 3 = (a + b)(a 2 - ab + b 2) 216x 3 + 64 = 8(3x + 2)(9x 2 - 6x + 4) 10x 3 - 80 10. 2x 4 + 7x 3 + 5x 2 10(x - 8) x 2(2x 2 + 7x + 5) 3 x 3 = (1x) 3 8 = (2) 3 2 2 10x - 80 = 10(x - 2)(x 2 + 2x + 4) 11. x + 10x + 16x x (x + 1)(2x + 5) 2 12. x + 9769 2 x(x + 10x + 16) 3 2 ( ) x (x(2x + 5) + 1(2x + 5)) ) 2 3 3 ( x 2 (2x 2 + 5x) + (2x + 5) 3 a - b = (a - b)(a + ab + b 3 © Houghton Mifflin Harcourt Publishing Company 9. Irreducible. ) x (x + 2x) + (8x + 16) 2 x(x(x + 2) + 8(x + 2)) Practice Explore Analyzing a Visual Model for Polynomial Factorization 2x 3 + 6x 3 a 3 + b 3 = (a + b)(a 2 - ab + b 2) 7. Concepts and Skills x(x + 2)(x + 3) 6. 3 ) x(x(x + 2) + 3(x + 2)) x 3 - 125 = (x - 5)(x 2 + 5x + 25) 8x 3 = (2x) ASSIGNMENT GUIDE x (x 2 + 2x) + (3x + 6) 3 a 3 - b 3 = (a - b)(a 2 + ab + b 2) 5. x 3 + 5x 2 + 6x x(x 2 + 5x + 6) 3 INTEGRATE TECHNOLOGY Emphasize that students should use caution when checking answers on a graphing calculator. The calculator provides support that the answer is correct, but it cannot be used to prove correctness. x(x + 2)(x + 8) Module 6 Lesson 4 316 Exercise A2_MNLESE385894_U3M06L4.indd 316 Depth of Knowledge (D.O.K.) Mathematical Practices 1–18 1 Recall of Information MP.5 Using Tools 19–22 2 Skills/Concepts MP.4 Modeling 23 1 Recall of Information MP.2 Reasoning 24 1 Recall of Information MP.3 Logic 25–26 2 Skills/Concepts MP.3 Logic 27–29 3 Strategic Thinking MP.2 Reasoning 3/19/14 1:37 PM Factoring Polynomials 316 Factor the polynomial by grouping. AVOID COMMON ERRORS 13. x 3 + 8x 2 + 6x + 48 x (x + 8) + 6(x + 8) 14. x3 + 4x 2 - x - 4 x 2(x + 4) - 1(x + 4) 2 Students may not recognize that a polynomial can sometimes be factored if they regroup the terms. Give students a pattern they can follow to test if factoring by grouping applies to a polynomial: first, rearrange the terms so that when they are grouped, they will have common factors; group the terms; factor each group, using factoring patterns if necessary; then, rearrange and assemble the factors using the distributive property (x 2 + 6)(x + 8) 15. 8x + 8x + 27x + 27 4 3 (x 2 - 1)(x + 4) (x - 1)(x + 1)(x + 4) 16. 27x 4 + 54x 3 - 64x - 128 8x 3(x + 1) + 27(x + 1) 27x 3(x + 2) - 64(x + 2) (8x 3 + 27)(x + 1) (2x + 3)(4x 2 - 6x + 9)(x + 1) 17. x 3 + 2x 2 + 3x +6 x (x + 2) + 3(x + 2) (27x 3 - 64)(x + 2) (3x - 4)(9x 2 + 12x + 16)(x + 2) 18. 4x 4 - 4x 3 - x + 1 4x 4 - 4x 3 - x + 1 2 (x 2 + 3)(x + 2) INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 After students have solved a polynomial 19. Engineering A new rectangular outbuilding for a farm is being designed. The outbuilding’s sides and bottom should be 4 feet thick. Its outer length should be twice its outer width and height. What should the outer dimensions of the outbuilding be if it is to have a volume of 2304 cubic feet? 2304 = (2x − 8)(x − 8)(x − 4) 2304 = 2x 3 - 32x 2 + 160x - 256 0 = 2x 3 - 32x 2 + 160x - 2560 0 = 2x 2(x - 16) + 160(x - 16) 0 = 2(x 2 + 80)(x - 16) The only real solution is x = 16. The outbuilding is 32 feet long, 16 feet wide, and 16 feet high. 20. Arts A piece of rectangular crafting supply is being cut for a new sculpture. You want its length to be 4 times its height and its width to be 2 times its height. If you want the wood to have a volume of 64 cubic centimeters, what will its length, width, and height be? V = (4x)(2x)(x) V = 8x 3 64 = 8x 3 8 = x3 2=x The length of the piece of crafting supply will be 8 cm, the width 4 cm, and the height 2 cm. Module 6 A2_MNLESE385894_U3M06L4 317 317 Lesson 6.4 (4x 3 - 1)(x - 1) Write and solve a polynomial equation for the situation described. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Alex Ramsay/Alamy equation using the zero-product property, help them understand and recall that the zeros of the polynomial function f(x) associated with the polynomial equation are the values of x where the graph of the polynomial function crosses the x-axis. The zeros of a function f(x) are also equivalent to the solutions of the equation f(x) = 0 and are related to the factors of the polynomial. 4x 3(x - 1) - 1(x - 1) 317 Lesson 4 6/8/15 11:42 PM 21. Engineering A new rectangular holding tank is being built. The tank’s sides and bottom should be 1 foot thick. Its outer length should be twice its outer width and height. INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Point out that students should review the What should the outer dimensions of the tank be if it is to have a volume of 36 cubic feet? 36 = (2x − 2)(x − 2)(x − 1) 36 = 2x 3 − 8x 2 + 10x − 4 factoring patterns they learned in Algebra 1 as well as the two useful factoring patterns in this lesson: 0 = 2x 3 − 8x 2 + 10x − 40 0 = 2x (x − 4) + 10(x − 4) 0 = 2(x 2 + 5)(x − 4) 2 Sum of two cubes: The only real solution is x = 4 . The tank is 8 feet long, 4 feet wide, and 4 feet high. a 3 + b 3 = (a + b)(a 2 - ab + b 2); 22. Construction A piece of granite is being cut for a building foundation. You want its length to be 8 times its height and its width to be 3 times its height. If you want the granite to have a volume of 648 cubic yards, what will its length, width, and height be? Difference of two cubes: a 3 - b 3 = (a - b)(a 2 + ab + b 2) V = (8x)(3x)(x) Emphasize that the exercises may go beyond factoring over the integers and may include factoring over the real numbers or complex numbers. V = 24x 3 648 = 24x 3 27 = x 3 3=x The length of the slab will be 24 yards, the width 9 yards, and the height 3 yards. AVOID COMMON ERRORS b. 3x 3 + 5 none c. 4x + 25 none 2 d. 27x 3 + 1000 sum of two cubes e. 64x - x + 1 none 3 2 © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gennadiy Iotkovskiy/Alamy 23. State which, if any, special factoring pattern each of the following polynomial functions follows: difference of two squares a. x 2 - 4 H.O.T. Focus on Higher Order Thinking 24. Communicate Mathematical Ideas What is the relationship between the degree of a polynomial and the degree of its factors? The degree of a polynomial is always at least 1 larger than the degree of any of its factors. Some students may not be sure of the point at which a polynomial has been factored completely. Discuss ways to determine whether the factoring is complete, checking that the greatest monomial factor has been factored out, and making sure that each factor is itself irreducible. 25. Critical Thinking Why is there no sum-of-two-squares factoring pattern? There is no sum-of-two-squares factoring pattern because any sum of two squares will only have complex roots as an answer. Module 6 A2_MNLESE385894_U3M06L4 318 318 Lesson 4 6/27/14 2:57 PM Factoring Polynomials 318 26. Explain the Error Jim was trying to factor a polynomial and produced the following result: PEER-TO-PEER DISCUSSION Instruct one student in each pair to write a polynomial while the other student gives verbal instructions for factoring the polynomial. Then have students switch roles, and repeat the exercise, giving instructions for factoring a different polynomial. 3x 3 + x 2 + 3x + 1 Write out the polynomial. 3x (x + 1) + 3(x + 1) Group by common factor. 2 3(x + 1)(x + 1) Regroup. 2 Explain Jim’s error. Jim misgrouped the polynomial. He should have grouped it like this: 3x 3 + x 2 + 3x + 1 Write out the polynomial. (x 2 + 1)(3x + 1) Regroup. 3x(x 2 + 1) + 1(x 2 + 1) JOURNAL Group by common factor. 27. Factoring can also be done over the complex numbers. This allows you to find all the roots of an equation, not just the real ones. Have students make a table describing the methods and patterns for factoring polynomials. Give examples of using the sum and difference of two cubes and of factoring polynomials by grouping. Complete the steps showing how to use a special factor identity to factor x 2 + 4 over the complex numbers. x2 + 4 Write out the polynomial. Rewrite as a difference of two squares. x - (-4) √ √ (x + -4 ) (x - -4 ) ―― 2 ―― Factor. (x + 2i) (x - 2i ) Simplify. 28. Find all the complex roots of the equation x 4 - 16 = 0. (x 2 + 4)(x 2 - 4) = 0 (x 2 + 4)(x - 2)(x + 2) = 0 Factors (x - 2) and (x + 2) will yield real roots. factor (x 2 + 4) will yield complex roots. © Houghton Mifflin Harcourt Publishing Company x2 + 4 x 2 -(-4) ―― (x + √―― -4 )(x - √-4 ) (x + 2i)(x - 2i) x + 2i = 0 or x = -2i x - 2i = 0 x = 2i The complex roots of the equation x 4 - 16 = 0 are x = -2i and x = 2i. 29. Factor x 3 + x 2 + x + 1 over the complex numbers. x3 + x2 + x + 1 x 2(x + 1) + 1(x + 1) (x + 1)(x 2 + 1) (x + 1)⎡⎣x 2 - (-1)⎤⎦ ―― ―― (x + 1)(x + √-1 )(x - √-1 ) (x + 1)(x + i)(x - i) Therefore, x 3 + x 2 + x + 1 = (x + 1)(x + i)(x - i) Module 6 A2_MNLESE385894_U3M06L4 319 319 Lesson 6.4 319 Lesson 4 6/8/15 11:42 PM Lesson Performance Task LANGUAGE SUPPORT Some students may not be familiar with the term flower bed. Explain that a bed can refer to an area for a garden. It may be enclosed by fencing, boards, or logs, and is sometimes built from the ground to a height of several inches to make it easier for the gardener to work in the soil. Have them look at the photo of the flower bed and describe any similarities they see between it and a bed that a person might sleep on. Both are rectangular, flat, and low to the ground. Sabrina is building a rectangular raised flower bed. The boards on the two shorter sides are 6 inches thick, and the boards on the two longer sides are 4 inches thick. Sabrina wants the outer length of her bed to be 4 times its height and the outer width to be 2 times its height. She also wants the boards to rise 4 inches above the level of the soil in the bed. What should the outer dimensions of the bed be if she wants it to hold 3136 cubic inches of soil? Let x = the external height Volume = Interior length(inches) · Interior Width(inches) · Interior Height (inches) 3136 = (4x - 12)(2x - 8)(x - 4) 3136 = 8x 3 - 88x 2 + 320x - 384 0 = 8x 3 - 88x 2 + 320x - 3520 0 = 8x 2(x - 11) + 320(x - 11) QUESTIONING STRATEGIES x = 11 is the only real solution Why is it important to move all the terms of the equation to one side? The other side will then be zero, and you can use the zero-product property to solve for the solution. 0 = (8x 2 + 320)(x - 11) Thus the external dimensions must be 44 inches long by 22 inches wide by 11 inches high. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gary K Smith/Alamy If the final factoring gives more than one solution, how can you determine the correct one? The answer must make sense for the problem. If the question is asking for a length of a board, then a negative answer would not be realistic. The answer must be positive. Module 6 320 Lesson 4 EXTENSION ACTIVITY A2_MNLESE385894_U3M06L4 320 Have students design a circular flower bed using a tire or other doughnut-shaped object. Have them discuss the constraints they would put on the dimensions of the bed and on the volume of the soil. Then ask them to set up a polynomial equation for the interior volume of the bed and solve for the unknown dimension. Have students discuss what shape would be best for a garden bed and why. A rectangular bed that is long and narrow allows the gardener to easily reach all plants without stepping on the soil. 6/27/14 2:59 PM Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Factoring Polynomials 320