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Unformatted text preview: 3x − 9 x − 6) − 5 = 2 3 2 x − ( 12 x − 9 x − 27 x − 18 ) − 5 3 Example Find ( x 5 + 4 x 4 + 72 x 2 − 8 x − 20 ) ÷ ( x + 6 ) . x 4 − 2 x 3 + 12 x 2 − 8 x + 6 x 5 + 4 x 4 + 0 x 3 + 72 x 2 − 8 x − 20 x5 + 6x4 − 2x4 + 0x3 − 2 x 4 − 12 x 3 12 x 3 + 72 x 2 12 x 3 + 72 x 2 − 8 x − 20 − 8 x − 48 28 The quotient function is q( x ) = x 4 − 2 x 3 + 12 x 2 − 8 and the remainder function is r ( x ) = 28 . We have that x 5 + 4 x 4 + 72 x 2 − 8 x − 20 = ( x + 6 ) ( x 4 − 2 x 3 + 12 x 2 − 8 ) + 28 . Consider the following. Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1320 5 4 2 72 Coefficients x + x − 8 x − 20 of x + 4 1 4 0 72 − 8 − 20 −6 1 12 − 72 0 48 −2 12 0 −8 −6 28 What do the numbers in the third row represent? 1 4 −6 0 12 −8 0 72 − 72 − 20 48 1 −2 12 0 −8 −6 28 Re mainder Coefficients of the quotient function starting with x 4 Thus, the quotient function is q( x ) = x 4 − 2 x 3 + 12 x 2 − 8 and the remainder function is r ( x ) = 28 . These are the same answers that we obtained above using long division. This process is called synthetic division. Synthetic division can only be used to divide a polynomial by another polynomial of degree one with a leading coefficient of one. Thus, you can NOT use synthetic division to find ( 12 x 4 − 17 x 3 − 21 x 2 + 7 ) ÷ ( 3 x − 2 ) . However, we can do the following division. 2 4 3 2 Example Use synthetic division to find ( 12 x − 17 x − 21 x + 7 ) ÷ x − . 3 4 3 2 Coefficients of 12x − 17 x − 21x + 7 12 − 17 − 21 0 7 − 18 − 12 12 −9 − 27 − 18 −5 8 −6 2 3 Coeff of quotient function starting with x 3 Thus, the quotient function is q( x ) = 12 x 3 − 9 x 2 − 27 x − 18 and the remainder function is r ( x ) = − 5 . These are the same answers that we obtained above using long division....
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This note was uploaded on 02/11/2014 for the course MATH 1320 taught by Professor Staff during the Spring '08 term at Toledo.

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