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3.2 Dividing Polynomials 11/28/2012

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Review: Quotient of Powers Ex. In general:

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Use Long Division Find the quotient 985 23. ÷ Divide 98 by 23. 985 23 -92 Subtract the product. 4 () 23 = 92 65 Bring down 5. Divide 65 by 23. 19 Remainder ANSWER The result is written as. 23 19 42 -46 Subtract the product. 2 () 23 = 46 42

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Example 1 Use Polynomial Long Division x 3x 3 +4x 24x 2 Subtract the product. () 4x + x 2x 2 = x 3x 3 4x 24x 2 + – 6x6xx 2x 2 – Bring down - 6x. Divide –x 2 by x – 4x4xx 2x 2 – Subtract the product. () 4x + x = x 2x 2 4x4x ––– – 2x2x – 4 Bring down - 4. Divide -2x by x 4 Remainder x 3x 3 + – 6x6x3x 23x 2 – 4x+4 x 3x 3 ÷x = x 2x 2 ANSWER The result is written as. x 2x 2 –– x2 x+4 4 + – 2x2x – 8 Subtract the product () 4x + 2 = 2x2x8 –––. x2x2 -x -2 - +

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Synthetic division: Is a method of dividing polynomials by an expression of the form x - k

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Example 1 Using Synthetic division x – (-4) in x – k form -4Coefficients of powers of x 1 3 -6 -4 k 1 -4 4 multiply -2 8 4 add coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. ANSWER x 2x 2 –– x2 x+4 4 + remainder

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k Isn’t this the remainder when we performed synthetic division? Remainder Thm:If a polynomials f(x) is divided by x – k, then the remainder is r = f(k)

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Example 2 Using Synthetic division and Remainder Theorem 3 Coefficients of powers of x 2 -7 0 6 -14 k 2 6 -3 multiply -3 -9 -23 add remainder -3 -9 P(3)= -23

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Example 3 Use Polynomial Long Division Can’t use synthetic division because it isn’t being divided by x-k - + - remainder

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Homework: Worksheet 3.2 #1-5all, 11-19odd, 23-25all

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