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Dividing Polynomials DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B LESSON6 . 5 Name Class Date Dividing Polynomials 6 . 5 Dividing Polynomials Essential Question: What are some ways to divide polynomials, and how do you know when Common Core Math Standards the divisor is a factor of the dividend? The student is expected to: Resource Locker COMMON CORE A-APR.D.6 Explore Evaluating a Polynomial Function Rewrite simple rational expressions in different forms; write a(x)/b(x) in Using Synthetic Substitution the form q(x) + r(x)/b(x), … using inspection, long division, … . Also A-APR.A.1, A-APR.B.3 Polynomials can be written in something called nested form. A polynomial in nested form is written in such a way that evaluating it involves an alternating sequence of additions and multiplications. For instance, the nested form of Mathematical Practices p x 4 x 3 3 x 2 2x 1 is p x x x 4x 3 2 1, which you evaluate by starting with 4, multiplying by ( ) = + + + ( ) = ( ( + ) + ) + COMMON the value of x, adding 3, multiplying by x, adding 2, multiplying by x, and adding 1. CORE MP.2 Reasoning Given p x 4 x3 3 x2 2x 1, find p 2 . Language Objective ( ) = + + + (- ) Rewrite p x as p x x x 4x 3 2 1. Work in small groups to complete a compare and contrast chart for ( ) ( ) = ( ( + ) + ) + dividing polynomials. Multiply. -2 · 4 = -8 Add. -8 + 3 = -5 Multiply. -5 · (-2) = 10 ENGAGE Add. 10 + 2 = 12 Multiply. 12 2 24 Essential Question: What are some · (- ) = - ways to divide polynomials, and how do Add. -24 + 1 = -23 you know when the divisor is a factor of You can set up an array of numbers that captures the sequence of multiplications and additions needed to find p( a). Using this array to find p( a) is called synthetic substitution. the dividend? 3 2 Given p( x) = 4 x + 3 x + 2x + 1, find p( -2) by using synthetic substitution. The dashed Possible answer: You can divide polynomials using arrow indicates bringing down, the diagonal arrows represent multiplication by –2, and the solid down arrows indicate adding. long division or, for a divisor of the form x a, - The first two steps are to bring down the leading number, 4, then multiply by the value you synthetic division. The divisor is a factor of the are evaluating at, -2. dividend when the remainder is 0. -2 4 3 2 1 -8 PREVIEW: LESSON 4 PERFORMANCE TASK Mifflin Houghton © Company Harcourt Publishing Add 3 and –8. View the Engage section online. Discuss the photo -2 4 3 2 1 and how the number of teams and the attendance are -8 both functions of time. Then preview the Lesson Performance Task. 4 -5 Module 6 through "File info" 321 Lesson 5 DO NOT EDIT--Changes must be made CorrectionKey=NL-B;CA-B Date Class Name n x r( x) 6 . 5 Dividing Polynomialsdivide polynomials, and how do you know________q( )whe+ , Resource x Locker ___a(x) in the form b( ) b( x) Essential Question: theWhat divisor are some is a factor ways toof the dividend?ifferent forms; write HARDCOVER PAGES 231240 A-APR.A.1, A-APR.B.3 COMMON CORE A-APR.D.6 Rewrite simple rational expressions in d … using inspection, long division, … Also Evaluating a Polynomial Function tten in such a way Explore Using Synthetic Substitution ce, the nested form of led nested form. A polynomial in nested form is wri ultiplying by ng sequence of additions and multiplications. For instan Polynomials can be written in something cal 2 + 1, which you evaluate by starting with 4, m x 4x + 3) + ) A2_MNLESE385894_U3M06L5 321 x = x ( 16/10/14 9:35 AM that evaluating it 2involves an1 is alternati p( ) ( 3 2x + g 2, multiplying by x, and adding 1. x + 3 x + p( x) = 4 . the value of x, adding 3, multiplying by x, addin -2) 2 1, find p( 3 3 x + 2x + x 4 x + 2 1. Turn to these pages to p( ) = 4x 3 + + Given x x x( + ) ) p x as p( ) = ( Rewrite ( ) 4 -8 -2 · = Multiply. 5 8 + 3 = - Add. - 10 find this lesson in the 5 · -2 = Multiply. - + 2 = 12 Add. 10 24 · -2 = - Multiply. 12 . 1 -23 nce of multiplications and on -24 + = hardcover student Add. (a) is called synthetic substituti f numbers that captures the seque The dashed a . Using this array to find p You can set up an array o ( ) and the 2) by using synthetict multiplication substitution. by 2, additions needed to find2 p (- – 3 2x + 1, find p x 3 x + p x) = 4 + n, the diagonal arrows represen Given ( arrow indicates bringing dow then multiply by the value you edition. solid down arrows indicate adding. ing down the leading number, 4, 1 2 The first two steps are to br 3 are evaluating at, 2. 4 - -2 -8 4 1 2 3 © Houghton Mifflin Harcourt Publishing Company Publishing Harcourt Mifflin Houghton © Add 3 and 8. 4 – -2 -8 Lesson 5 5 4 - 321 6/27/14 7:54 PM Module 6 A2_MNLESE385894_U3M06L5 321 321 Lesson 6 . 5 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through "File info" DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B CorrectionKey=NL-B;CA-B Multiply the previous answer by –2. Continue this sequence of steps until you reach the last addition. -2 4 3 2 1 EXPLORE -2 4 3 2 1 -8 10 Evaluating a Polynomial Function -8 10 -24 4 -5 Using Synthetic Substitution 4 -5 12 -23 p( -2) = -23 INTEGRATE TECHNOLOGY Reflect Students have the option of completing the 1. Discussion After the final addition, what does this sum correspond to? polynomial division activity either in the book The final sum represents the value of p (x) where x = -2. or online. Explain 1 Dividing Polynomials Using Long Division QUESTIONING STRATEGIES Recall that arithmetic long division proceeds as follows. What operation are you doing and what are Divisor 23 ← Quotient you finding when you use synthetic 12 ⟌–––– 277 ← Dividend 24 substitution on the polynomial function p( x) to find ― 37 p( a) ? You are dividing the polynomial function p( x) 36 by the quantity x a , and you are finding the ― ( - ) 1 ← Remainder value of p( a) . dividend remainder Notice that the long division leads to the result _______ quotient ________ . Using the divisor = + divisor 277 1 ___ __ numbers from above, the arithmetic long division leads to 12 = 23 + 12 . Multiplying through by the divisor yields the result dividend = ( divisor)( quotient) + remainder. (This can be used as a means of checking your work.) Mifflin Houghton © Company Harcourt Publishing EXPLAIN 1 Example 1 Given a polynomial divisor and dividend, use long division to find the quotient and remainder. Write the result in the form Dividing Polynomials Using Long dividend = (divisor) (quotient) + remainder and then carry out the Division multiplication and addition as a check. 3 2 2 4 x + 2 x + 3x + 5 ÷ x + 3x + 1 ( ) ( ) AVOID COMMON ERRORS Begin by writing the dividend in standard form, including terms with a coefficient of 0 (if any). Students may have difficulty relating the familiar 3 2 4 x + 2 x + 3x + 5 long-division process for whole numbers to Write division in the same way as you would when dividing numbers. identifying the process for polynomials using the x 2 3x 1 4 x 3 2 x2 3x 5 algorithm for finding dividend = (divisor)(quotient) + + ⟌––––––––––––––––– + + + + remainder. Point out that polynomial long division leads to this result: ________ dividend quotient _________ remainder , divisor = + divisor Module 6 322 Lesson 5 which is equivalent to dividend = (divisor)(quotient) + remainder if you multiply each term by the divisor. PROFESSIONAL DEVELOPMENT Showing an example of arithmetic long division A2_MNLESE385894_U3M06L5 322 6/27/14 7:01 PM Math Background alongside an example of polynomial division may help students make the connection. Division of polynomials is related to division of whole numbers. Given _____P( x) _____R (x) polynomials P( x) and D( x) , where D( x) ≠ 0, we can write = Q( x) + , D( x) D( x) where the remainder R( x) is a polynomial whose degree is less than that of D( x) . (If the degree of R( x) were not less than the degree of D( x) , we would be able to continue dividing.) Equivalently, P (x) = Q( x) D( x) + R( x) . This last expression can be used to justify the Remainder Theorem. Notice that when D( x) is a linear divisor of the form x - a, the expression becomes P( x) = Q( x)( x - a) + r, where the remainder r is a real number. Dividing Polynomials 322 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B Find the value you need to multiply the divisor by so that the first term matches 2 with the first term of the dividend. In this case, in order to get 4x , we must multiply QUESTIONING STRATEGIES 2 x by 4x. This will be the first term of the quotient. 4x How can you tell if you are finished solving a 2 3 2 x + 3x + 1 ⟌––––––––––––––––– 4 x + 2 x + 3x + 5 polynomial division problem? The remainder Next, multiply the divisor through by the term of the quotient you just found and has a degree less than the degree of the divisor, or 2 3 2 subtract that value from the dividend.
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