6.2 Multiplying Polynomials

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1 Locker LESSON 6. Multiplying Polynomials PAGE 7 BEGINS HERE Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? Common Core Math Standards The student is epected to: A-APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Also A-APR., F-BF.1b Mathematical Practices MP. Modeling Language Objective Work in pairs to complete a compare and contrast chart for adding/ subtracting and multiplying polynomials. ENGAGE Essential Question: How do you multiply polynomials, and what type of epression is the result? Possible answer: You multiply two polynomials by multiplying each term of one polynomial with each term of the other polynomial. The product is another polynomial. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the total amount of oil produced is a function of both the number of wells and the amount produced by each well. Then preview the Lesson Performance Task. Eplore Analyzing a Visual Model for Polynomial Multiplication The volume of a rectangular prism is the product of the length, width, and height of that prism. If the dimensions are all known, then the volume is a simple calculation. What if some of the dimensions are given as binomials? A binomial is a polynomial with two terms. How would you find the volume of a rectangular prism that is + units long, + units wide, and units high? The images below show two methods for finding the solution. V = length width height = ( + ) ( + ) + + The first model shows the rectangular prism, and its volume is calculated directly as the product of two binomials and a monomial. The second image divides the rectangular prism into four smaller prisms, the dimensions of which are each monomials. The volume of a cube ( V 1 ) where all sides have a length of, is =. The volume of a rectangular prism ( V ) with dimensions by by is =. The volume of a rectangular prism ( V ) with dimensions by by is =. The volume of a rectangular prism ( V ) with dimensions by by is = 6. v = volume of v 1 = volume of So the volume of the rectangular prism is the sum of the volumes of the four smaller regions. V 1 + V + V + V = = v = v 1 + v + v + v v = volume of v = volume of Resource Locker Module 6 7 Lesson Name Class Date 6. Multiplying Polynomials Essential Question: How do you multiply polynomials, and what type of epression is the result? A-APR.1 For the full tet of this standard, see the table starting on page CA. Also A-APR., F-BF.1b Eplore Analyzing a Visual Model for Polynomial Multiplication The volume of a rectangular prism is the product of the length, width, and height of that prism. If the dimensions are all known, then the volume is a simple calculation. What if some of the dimensions are given as binomials? A binomial is a polynomial with two terms. How would you find the volume of a rectangular prism that is + units long, + units wide, and units high? The images below show two methods for finding the solution. V = length width height = ( + ) ( + ) v = v 1 + v + v + v v = volume of v = volume of + v 1 = volume of v = volume of + The first model shows the rectangular prism, and its volume is calculated directly as the product of two binomials and a monomial. The second image divides the rectangular prism into smaller prisms, the dimensions of which are each. The volume of a cube ( V 1 ) where all sides have a length of, The volume of a rectangular prism ( V ) with dimensions by by The volume of a rectangular prism ( V ) with dimensions by by The volume of a rectangular prism ( V ) with dimensions by by = is. = is. = is. = 6 four monomials is. So the volume of the rectangular prism is the sum of the volumes of the four smaller regions. V 1 + V + V + V = = Resource Module 6 7 Lesson HARDCOVER PAGES 7 6 Turn to these pages to find this lesson in the hardcover student edition. 7 Lesson 6.

2 Reflect 1. If all three dimensions were binomials, how many regions would the rectangular prism be divided into? 8. Discussion Can this method be applied to finding the volume of other simple solids? Are there solids that this process would be difficult to apply to? Are there any solids that this method cannot be applied to? Yes. It would be difficult to find the volume of any shape, such as a pyramid, that did not subdivide into smaller iterations of that shape and did not stack together well. It cannot be applied, for eample, to a sphere. Eplain 1 Multiplying Polynomials Multiplying polynomials involves using the product rule for eponents and the distributive property. The product of two monomials is the product of the coefficients and the sum of the eponents of each variable. 6 = y z y z = -10 y + z = 0 = -10 y 6 z When multiplying two binomials, the distributive property is used. Each term of one polynomial must be multiplied by each term of the other. ( + ) (1 + ) = (1 + ) + ( + 1) = (1) + () + () + (1) = = + + PAGE 8 BEGINS HERE EXPLORE Analyzing a Visual Model for Polynomial Multiplication INTEGRATE TECHNOLOGY Students have the option of completing the polynomial multiplication activity either in the book or online. QUESTIONING STRATEGIES How does a polynomial model the volume of a real-world figure like a rectangular prism with variable dimensions? The volume of a figure like a rectangular prism can be written as a cubic polynomial because the polynomial represents three dimensions. For the rectangular prism, the cubic may be a product of the dimensions. The polynomial + + is called a trinomial because it has three terms. Eample 1 Perform the following polynomial multiplications. ( + ) (1 - + ) Find the product by multiplying horizontally. ( + ) ( - + 1) Write the polynomials in standard form. ( ) + (-) + (1) + ( ) + (-) + (1) Distribute the and the Simplify Combine like terms. Therefore, ( + ) ( - + 1) = Module 6 8 Lesson PROFESSIONAL DEVELOPMENT Math Background In the previous lesson, students discovered that polynomials are closed under addition and subtraction. In this lesson, students learn that polynomials are also closed under multiplication; the product of two polynomials is a polynomial. When we multiply two nonzero integers p by q, we say that the product is an integer pq such that each digit of q is multiplied by each digit of p, and then the partial products are added. Multiplication of polynomials is much the same. Given polynomials P () and Q (), where both P () and Q () 0, we can write P () Q () = R (). R () will be a simplified polynomial with like terms combined. EXPLAIN 1 Multiplying Polynomials AVOID COMMON ERRORS Students often are unsure whether to multiply polynomials horizontally or vertically. Point out that if the polynomials have many terms, multiplying them vertically may prevent errors because vertical multiplication is familiar, and the locations of the product terms are similar to place value in a numerical product. If the polynomials have few terms, multiplying horizontally may be more convenient, as long as the student remembers to use the distributive property to multiply each term of one polynomial by all other terms of the other polynomial. Multiplying Polynomials 8

3 QUESTIONING STRATEGIES Is the commutative property of multiplication true for the multiplication of polynomials? Eplain. Yes, the product will be the same regardless of the order in which polynomials are multiplied. After you have multiplied two polynomials, how can you make sure you have not missed any terms in the process? Before simplifying, the product of a polynomial with m terms and a polynomial with n terms has mn terms, so count the number of terms in the product. EXPLAIN Modeling with Polynomial Multiplication INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Point out that using a table with color to organize the products may be helpful when finding the product of real-world polynomials. For eample, to find ( + - ) ( - + 1), the table below might be used, with the terms of each trinomial either above the columns, or alongside the rows Like terms, shown with the same color, are combined to complete the product. PAGE 9 BEGINS HERE ( - ) ( ) Find the product by multiplying vertically Write each polynomial in standard form Multipy - and ( ) Multipy and ( ) Combine like terms. Therefore, ( - ) ( ) = Your Turn. ( + ) ( ). ( - 6) ( ) Eplain Modeling with Polynomial Multiplication Many real-world situations can be modeled with polynomial functions. Sometimes, a situation will arise in which a model is needed that combines two quantities modeled by polynomial functions. In this case, the desired model would be the product of the two known models. Eample 8 ( ) + (-7) + () + ( ) + (-7) + () Distribute the and the Simplify Combine like terms. (- ) + (-8) + () - 6 (- ) - 6 (-8) - 6 () Distribute the and the Simplify Combine like terms. Find the polynomial function modeling the desired relationship. Mr. Silva manages a manufacturing plant. From 1990 through 00, the number of units produced (in thousands) can be modeled by N () = , where is the number of years since The average cost per unit (in dollars) can be modeled by C () = , where is the number of years since Write a polynomial T () that can be used to model Mr. Silva s total manufacturing cost for those years. The total manufacturing cost is the product of the number of units made and the cost per unit. T () = N () C () Module 6 9 Lesson COLLABORATIVE LEARNING Small Group Activity Have groups of students describe how to multiply polynomials. Ask them to write an eample multiplication problem in a graphic organizer similar to the one shown. Students then pass the organizer to another student, who writes in the net step and describes it. They continue to pass the organizers until each problem is solved and all steps are eplained. A Sample organizer is shown Lesson 6.

4 Multiply the two polynomials QUESTIONING STRATEGIES What property of eponents is used to find the partial products? When you multiply two powers with the same base, you add their eponents. B Therefore, the total manufacturing cost can be modeled by the following polynomial, where is the number of years since T () = Ms. Liao runs a small dress company. From 199 through 00, the number of dresses she made can be modeled by N () = , and the average cost to make each dress can be modeled by C () = , where is the number of years since 199. Write a polynomial that can be used to model Ms. Liao s total dressmaking costs, T (), for those years. The total dressmaking cost is the product of the number of dresses made and the cost per dress. T () = N () C () Multiply the two polynomials PAGE 60 BEGINS HERE Therefore, the total dressmaking cost can be modeled by the following polynomial, where is the number of years since 199. T () = Module 6 0 Lesson DIFFERENTIATE INSTRUCTION Multiple Representations Have students work in small groups to multiply two polynomials, such as ( + - ) ( - + 1). Each student in the group should choose a different method, such as multiplying horizontally, multiplying vertically, or using a table. Have students discuss the ways in which the methods are alike and the ways in which they differ. Multiplying Polynomials 0

5 EXPLAIN Verifying Polynomial Identities Your Turn. Brent runs a small toy store specializing in wooden toys. From 000 through 01, the number of toys Brent made can be modeled by N () = , and the average cost to make each toy can be modeled by C () = , where is the number of years since 000. Write a polynomial that can be used to model Brent s total cost for making the toys, T (), for those years. AVOID COMMON ERRORS Students may think that they need to analyze each side of a polynomial equation in order to verify that the equation epresses a polynomial identity. Point out that if one side of the equation is a monomial, then that side is complete. There may be more than one way to proceed, but the arithmetic operations must be performed on both sides, if necessary, until the two sides match. The total cost is the product of the number of toys made and the cost per toy. Multiply the two polynomials Therefore, the total cost of making the toys can be modeled by the following polynomial, where is the number of years since 000. T () = QUESTIONING STRATEGIES How do you verify a polynomial identity? You perform the operations indicated on each side of the identity until the two sides match. Eplain Verifying Polynomial Identities You have already seen certain special polynomial relationships. For eample, a difference of two squares can be easily factored: - a = ( + a) ( - a). This equation is an eample of a polynomial identity, a mathematical relationship equating one polynomial quantity to another. Another eample of a polynomial identity is ( + a) - ( - a) = a. The identity can be verified by simplifying one side of the equation to match the other. Eample Verify the given polynomial identity. ( + a) - ( - a) = a The right side of the identity is already fully simplified. Simplify the left-hand side. ( + a) - ( - a) = a + a + a -( - a + a ) = a Square each binomial. + a + a - + a - a = a Distribute the negative. - + a + a + a - a = a Rearrange terms. a = a Simplify. Therefore, ( + a) - ( - a) = a is a true statement. Module 6 1 Lesson LANGUAGE SUPPORT Communicate Math Have students complete a chart like the following showing similarities and differences: Operation Alike Different Add and Subtract Polynomials The result is another polynomial. You can only add and subtract like terms. Multiply Polynomials The result is another polynomial. You don t need to multiply like terms. 1 Lesson 6.

6 (a + b) ( a - ab + b ) = a + b The right side of the identity is already fully simplified. Simplify the left-hand side. Your Turn (a + b) ( a - ab + b ) = a + b a ( a ) + a ( ) + a ( b ) + b ( a ) + (-ab) + b ( b ) = a + b Distribute a and b. a - a b + ab + a b - ab + = a + b a - + a b + a b a b - ab + b = a + b Rearrange terms. Therefore, (a + b) ( a - ab + b ) = a + b is a 6. Show that a - b = (a - b) ( a + a b + a b + ab + b ). 7. Show that (a - b) ( a + ab + b ) = a - b. Eplain a + b = a + b Combine like terms. statement. The left side of the identity is already fully simplified. Simplify the right-hand side. a - b = a (a ) + a (a b) + a (a b ) + a (ab ) + a (b ) - b (a ) - b (a b) - b (a b ) - b (ab ) - b (b ) a - b = a + a b + a b + a b + ab - a b - a b - a b - ab - b a - b = a - b The right side of the identity is already fully simplified. Simplify the left-hand side. a (a ) + a (ab) + a (b ) - b (a ) - b (ab) - b (b ) = a - b Distribute a and b. a + a b + ab - a b - ab - b = a - b Simplify. a - b = a - b Combine like terms. Using Polynomial Identities The most obvious use for polynomial identities is simplifying algebraic epressions, but polynomial identities often turn out to have nonintuitive uses as well. Eample -ab For each situation, find the solution using the given polynomial identity. The polynomial identity ( + y ) = ( - y ) + (y) can be used to identify Pythagorean triples. Generate a Pythagorean triple using = and y =. Substitute the given values into the identity. ( + ) = ( - ) + ( ) (16 + 9) = (16-9) + () () = (7) + () 6 = = 6 Therefore, 7,, is a Pythagorean triple. b true b Simplify. PAGE 61 BEGINS HERE EXPLAIN Using Polynomial Identities CONNECT VOCABULARY Students may not understand identity in the contet of using polynomials identities. Tell them that once an identity is established, they should then apply the identity to numbers, much as they would apply a known formula to a geometric figure. In the process of using the identity, they do not re-verify the identity. QUESTIONING STRATEGIES How are polynomial identities used? They may be used to simplify algebraic epressions or to find shortcuts for polynomial-based formulas or mental math calculations. Module 6 Lesson Multiplying Polynomials

7 AVOID COMMON ERRORS Regardless of the method students use to multiply polynomials, a common error is to use the properties of eponents incorrectly, multiplying eponents that should be added. Remind students that the product of two powers with the same base is the base raised to the sum of the powers, or b m b n = b m + n. PAGE 6 BEGINS HERE B The identity ( + y) = + y + y can be used for mental-math calculations to quickly square numbers. Find the square of 7. Find two numbers whose sum is equal to 7. Let = 0 and y = 7 Evaluate ( 0 + ) 7 = = = 79 Verify by using a calculator to find 7. 7 = 79 Your Turn 8. The identity ( + y) ( - y) = - y can be used for mental-math calculations to quickly multiply two numbers in specific situations. Find the product of 7 and. (Hint: What values should you choose for and y so the equation calculates the product of 7 and?) Substitute = 0 and y = into the identity and evaluate. (0 + ) (0 - ) = 0-7 = = The identity ( - y) = - y + y can also be used for mental-math calculations to quickly square numbers. Find the square of 18. (Hint: What values should you choose for and y so the equation calculates the square of 18?) - 7 = 18 Substitute = and y = 7 into the identity and evaluate. ( - 7) = = = 6-01 = Module 6 Lesson Lesson 6.

8 Elaborate 10. What property is employed in the process of polynomial multiplication? The distributive property. 11. How can you use unit analysis to justify multiplying two polynomial models of real-world quantities? The units of the polynomials need to combine in such a way that their product is the desired unit. 1. Give an eample of a polynomial identity and how it s useful. See student work; answers will vary. ELABORATE QUESTIONING STRATEGIES How is the distributive property used to multiply two polynomials? Each monomial term of one polynomial must be multiplied by the each term of other polynomial, so the distributive property applies. 1. Essential Question Check-In When multiplying polynomials, what type of epression is the product? A polynomial CONNECT VOCABULARY Relate the prefies bi- and tri- to the meaning of binomial (two terms) and trinomial (three terms). Evaluate: Homework and Practice 1. The dimensions for a rectangular prism are + for the length, + 1 for the width, and for the height. What is the volume of the prism? ( + ) ( + 1) = Perform the following polynomial multiplications.. ( - ) ( + - 1) Online Homework Hints and Help Etra Practice SUMMARIZE THE LESSON What points should you remember when multiplying polynomials? Use the distributive property to multiply every term of one polynomial by every term of the other polynomial, combine like terms, and align like terms. ( ) + () + (-1) - ( ) - () - (-1) Distribute the and the Simplify Combine like terms.. ( + + 1) ( ) Multiply - and ( + + 1) Multiply 6 and ( + + 1) Multiply and ( + + 1) Combine like terms. Module 6 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1 1 Recall of Information MP.6 Precision 7 1 Recall of Information MP.6 Precision 8 11 Skills/Concepts MP. Modeling 1 1 Skills/Concepts MP. Reasoning Strategic Thinking MP. Reasoning EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Analyzing a Visual Model for Polynomial Multiplication Eample 1 Multiplying Polynomials Eample Modeling with Polynomial Multiplication Eample Verifying Polynomial Identities Eample Using Polynomial Identities Practice Eercise 1 Eercises 7 Eercises 8 11 Eercises 1 1 Eercises 16 1 Strategic Thinking MP. Using Tools Multiplying Polynomials

9 INTEGRATE TECHNOLOGY Point out that when multiplying two polynomials in one variable, students can graph the epressions before they are multiplied and, again, after they are multiplied. The graphs should be coincident.. ( ) ( ). ( + y) ( - y + y ) Multiply and ( ) Multiply 9 and ( ) Multiply and ( ) Combine like terms. ( ) + (-y) + ( y ) + y ( ) + y (-y) + y ( y ) Distribute the and the y. 6-8 y + y + 1 y - 0 y + 10 y Simplify y - 16 y + 10 y Combine like terms. 6. ( + + 1) ( - - ) Multiply - and ( + + 1) Multiply - and ( + + 1). + Multiply and ( + + 1) Combine like terms. 7. ( + + ) ( + - 1) Multiply -1 and ( + + ) Multiply and ( + + ) Multiply and ( + + ) Combine like terms. Module 6 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices Strategic Thinking MP. Reasoning Strategic Thinking MP. Logic 6 Strategic Thinking MP. Reasoning Lesson 6.

10 Write a polynomial function to represent the new value. 8. The volume of a stock, or number of shares traded, is modeled over time during a given day by S () = The cost per share of that stock during that day is modeled by C () = Write a polynomial function V () to model the changing value during that day of the trades made of shares of that stock. The value is equal to the number of shares traded times the cost per share. V () = ( ) ( ) V () = A businessman models the number of items (in thousands) that his company sold from 1998 through 00 as N () = and the average price per item (in dollars) as P () = 0. +, where represents the number of years since Write a polynomial R () that can be used to model the total revenue for this company. The total revenue will be the product of the number of items sold and the price each item is sold at. PAGE 6 BEGINS HERE INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP. Help students clarify how to multiply polynomials by having them work in groups. Have one student complete one step of the multiplication process, including an eplanation of the process, then pass the problem to another student, who completes the second step, including an eplanation. Continue passing the problem until it is complete. Multiply the two polynomials. R () = (0. + ) ( ) = 0. (-0.1 ) + 0. ( ) + 0. (-) + 0. () + (-0.1 ) + ( ) + (-) + () = = Biology A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial b (y) = y + y where y is the number of years after the tree reaches a height of 6 feet. The number of leaves on each branch can be modeled by the polynomial l (y) = y + y + y. Write a polynomial describing the total number of leaves on the tree. T (y) = ( y + y) ( y + y + y) = 8 y + 1 y + y + y + y + y = 8 y + 1 y + 7 y + y Module 6 6 Lesson Multiplying Polynomials 6

11 MULTIPLE REPRESENTATIONS To help students structure how to multiply polynomials, have them use tables similar to the one shown below for ( + + 1) ( + - ). These diagrams provide more visual support than the more standard vertical method. Have students share their tables with a partner, describing the patterns they see and telling how they got their product polynomials Physics An object thrown in the air has a velocity after t seconds that can be described by v (t) = -9.8t + (in meters/second) and a height h (t) = -.9 t + t + 60 (in meters). The object has mass m = kilograms. The kinetic energy of the object is given by K = 1 m v, and the potential energy is given by U = 9.8mh. Find an epression for the total kinetic and potential energy K + U as a function of time. What does this epression tell you about the energy of the falling object? K = 1_ ()(-9.8t + ) K = (-9.8t + ) K = 96.0 t - 70.t + 76 U = 9.8 () (-.9 t + t + 60) U = t + 70.t K + U = (96.0 t - 70.t + 76) + (-96.0 t + 70.t ) = 17 Since the sum is a constant, this means that the energy of the object is constant and that as it gains kinetic energy by falling, it loses the same amount of potential energy. Verify the given polynomial identity. 1. ( + y + z) = + y + z + y + z + yz The right side of the identity is already fully simplified. Simplify the left-hand side. ( + y + z) = () + (y) + (z) + y () + y (y) + y (z) + z () + z (y) + z (z) = + y + z + y + y + yz + z + zy + z = + y + z + y + y + z + z + yz + zy = + y + z + y + z + yz 1. a + b = (a + b) ( a - a b + a b - a b + b ) The left side of the identity is already fully simplified. Simplify the right-hand side. a - a b + a b - a b + b a + b a b - a b + a b - a b + b a - a b + a b - a b + a b a + b 1. - y = ( - y) ( + y) ( + y ) The left side of the identity is already fully simplified. Simplify the right-hand side. Eamine ( - y) ( + y) ( + y ). Recall that ( + y) ( - y) = - y. Substitute on the right side of the equation. - y = ( - y ) ( + y ) - y = ( ) + ( y ) - y ( ) - y ( y ) - y = - y Module 6 7 Lesson 7 Lesson 6.

12 1. (a + b )( + y ) = (a - by) + (b + ay) INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. When students multiply polynomials, they a ( ) + a (y ) + b ( ) + b (y ) = (a) - (a)(by) + (by) + (b) + (b)(ay) + (ay) a + a y + b + b y = a - aby + b y + b + aby + a y a + a y + b + b y = a + a y + b + b y may leave out some of the partial products. Tell students to write down all of the partial products and circle the monomial in each one. Then ask them to draw arrows from the monomial in the first polynomial to the matching monomial in the second polynomial. Find the square of the number or the product of the numbers using one or more of these identities. ( + y) = + y + y, ( + y)( - y) = - y, or ( - y) = - y + y. 16. = (0 + ) 17. = (0 + ) = = = 189 = = (90-1) = (0 - ) = = 791 When using the rules for some special products of polynomials, students often forget to apply the power of a power property of eponents to the coefficients of terms in the polynomial. Suggest that students first write the coefficient and variable within parentheses, with the eponent applied to both, and then simplify. = = (6 + 8)(6-8) = 6-8 Avoid Common Errors = = (6 + )(6 - ) = 6 - = (60 + ) - = (0 - ) - 6 = = = = = 0 = 9 Each term of is a perfect cube. Let a = and b =. Then: = - ( ) ( = ( - ) + (- ) + ( - ) = ( - )( ) ) The student substituted - for b instead of for b in the ab and b terms of the trinomial = - ( ) ( = ( - ) + ( ) + ( ) = ( - )( ) Module 6 IN_MNLESE8988_UM06L.indd 8. Eplain the Error A polynomial identity for the difference of two cubes is a - b = (a - b)(a + ab + b ). A student uses the identity to factor Identity the error the student made, and then correct it. ) 8 Lesson 8/0/17 8:0 AM Multiplying Polynomials 8

13 PEER-TO-PEER DISCUSSION Instruct one student in each pair to write two polynomials in standard form while the other student gives verbal instructions for multiplying the polynomials. Then have students switch roles, repeat the eercise, and give instructions for multiplying two new polynomials. JOURNAL Have students make a table describing the methods for multiplying polynomials. Give eamples for multiplying monomials, binomials, and trinomials, as well as for verifying polynomial identities. PAGE 6 BEGINS HERE. Determine how many terms there will be after performing the polynomial multiplication. a. ()() X 1 b. ()( + 1) 1 X c. ( + 1) ( - 1) 1 X d. ( + ) ( - + 1) 1 X a. () () = 1 1 term b. () ( + 1) = 6 + terms c. ( + 1) ( - 1) = - 1 terms d. ( + ) ( - + 1) = terms H.O.T. Focus on Higher Order Thinking. Multi-Step Given the polynomial identity: 6 + y 6 = ( + y ) ( - y + y ) a. Verify directly by epanding the right hand side. 6 + y 6 = ( + y ) ( - y + y ) 6 + y 6 = ( ) + (- y ) + ( y ) + y ( ) + y (- y ) + y ( y ) 6 + y 6 = 6 - y + y + y - y + y y 6 = 6 + y - y + y - y + y y 6 = 6 + y 6 b. Use another polynomial identity to verify this identity. (Note that a 6 = ( a ) = ( a ) ) Use a 6 = ( a ) to replace 6 + y 6 with ( ) + ( y ). Now use the identity for the sum of two cubes, (a + b) ( a - ab + b ) = a + b to simplify ( ) + ( y ). ( ) + ( y ) = ( + y ) ( ( ) - ( ) ( y ) + ( y ) ) = ( + y ) ( - y + y ). Communicate Mathematical Ideas Eplain why the set of polynomials is closed under multiplication. Since a m b n = ab m + n for real numbers a and b and whole numbers m and n, the product of two monomials is another monomial. Therefore, the product of two polynomials, which are sums of monomials, is again a sum of monomials, which is another polynomial. 6. Critical Thinking Eplain why every other term of the polynomial product ( - y) written in standard form is subtracted when ( - y) is raised to the fifth power. When the power of y in a term of the product is odd the term is subtracted, and when the power of y in a term of the product is even the term is added. Module 6 9 Lesson 9 Lesson 6.

14 Lesson Performance Task The table presents data about oil wells in the state of Oklahoma from 199 through 008. Year Number of Wells Average Daily Oil Production per Well (Barrels) 008 8, , , , , , , ,160. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP. Ask students to look at the data table and describe any trends they see in number of wells and daily output over time. Both decrease with time. Based on these trends, have students predict the behavior of the total daily oil output over time. It will decrease. Have students graph the total daily output function D (t) for the time period from 199 to 008 to test their predictions , , , , , , , , ,19.99 a. Given the data in this table, use regression to find models for the number of producing wells (cubic regression) and average daily well output (quadratic regression) in terms of t years since 199. b. Find a function modeling the total daily oil output for the state of Oklahoma. a. Producing oil wells: W (t) = 0.88 t t - 110t Average daily output (in barrels per well): O (t) = t - 0.1t +.01 where t equals time (199 = 1, 199 =, etc.) b t +9.6 t -110t t -0.1t t t -.1t t -.9 t t t AVOID COMMON ERRORS Some students may multiply eponents instead of adding them. Polynomials have two types of numerical values: eponents and coefficients. In the term 0.88 t, the coefficient is 0.88 and the eponent is. Have students eplain what to do with each value when multiplying two polynomials. Multiply the coefficients and add the eponents t t t t t t t t t +807 To the correct number of significant figures, D (t) = t t t t t Module 6 0 Lesson EXTENSION ACTIVITY Have students research the price of oil per barrel for each year from 199 to 008 and use polynomial regression to find a model P (t) for the price of oil. Then have students use the model and the daily output model O (t) they determined in the Performance Task to find a function modeling the total value of oil produced each day by the state of Oklahoma. Ask students to calculate the daily oil income in 199 and compare it to 008. Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Multiplying Polynomials 0