Where q x x 4 2 x 3 12 x 2 8 f x x 6

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Unformatted text preview: Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1320 Example If f ( x ) = x 5 + 4 x 4 + 72 x 2 − 8 x − 20 , then find f ( − 6 ) . f ( − 6 ) = ( − 6 ) 5 + 4 ( − 6 ) 4 + 72 ( − 6 ) 2 − 8 ( − 6 ) − 20 = − 7776 + 4 ( 1296 ) + 72 ( 36 ) − 8 ( − 6 ) − 20 = − 7776 + 5184 + 2592 + 48 − 20 = 28 This calculation would have been faster (and easier) using the fact that x 5 + 4 x 4 + 72 x 2 − 8 x − 20 = ( x + 6 ) ( x 4 − 2 x 3 + 12 x 2 − 8 ) + 28 that we obtained in the example above. where q( x ) = x 4 − 2 x 3 + 12 x 2 − 8 . f ( x ) = ( x + 6 ) q ( x ) + 28 , Thus, Thus, f ( − 6 ) = ( − 6 + 6 ) q( − 6 ) + 28 = 0 ⋅ q( − 6 ) + 28 = 0 + 28 = 28 . This result can be explained by the following theorem. Theorem (The Remainder Theorem) Let p be a polynomial. If p( x ) is divided by x − a , then the remainder is p( a ) . Proof If p( x ) is divided by x − a , then p( x ) = ( x − a ) q( x ) + r ( x ) . Thus, p( a ) = ( a − a ) q( a ) + r ( a ) = 0 ⋅ q( a ) + r ( a ) = 0 + r ( a ) = r ( a ) . Example If g ( x ) = 12 x 4 − 17 x 3 − 21 x 2 + 7 , then find g 2 . 3 4 3 2 Using synthetic division to find ( 12 x − 17 x − 21 x + 7 ) ÷ x − that 4 3 2 Coefficien tsof 12x − 17 x − 21x + 7 12 − 17 − 21 0 7 8 12 −6 − 18 − 12 −9 − 27 − 18 −5 Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~anderson/1320 2 3 2 , we have 3 2 Thus, the remainder is − 5 . Thus, g = − 5 . 3 Example If h( x ) = x 5 − 8 x 4 + 12 x 3 + 23 x 2 − 16 x − 37 , then find h( 3 ) . Using synthetic division to find...
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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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