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mac1105_lecture19_2 - L19 Zeros of a Polynomial Function...

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200 L19 Zeros of a Polynomial Function Division Algorithm for Polynomials If f and g are two polynomials and g is not the zero polynomial, then there are the unique polynomials q (quotient) and r (remainder) such that ( ) ( ) ( ) ( ) ( ) f x r x q x g x g x = + or ( ) ( ) ( ) ( ) f x q x g x r x = + where ( ) r x either the zero polynomial or of degree less than the degree of ( ) g x . Note : If ( ) g x x c = , then ( ) ( )( ) f x q x x c r = + , where r is a number. If x c = , then ( ) f c = Remainder Theorem If a polynomial ( ) f x is divided by ( ) x c , then the remainder ( ) r f c = . 201 Example : Find the remainder if 4 3 ( ) 6 2 f x x x = + is divided by ( 2) x + . Use synthetic division the Remainder Theorem Example : Use the Remainder Theorem to find (3) f if 6 5 2 ( ) 6 54 16 1 f x x x x x = + +
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