200 L19 Zeros of a Polynomial FunctionDivision Algorithm for PolynomialsIf fand gare two polynomials and gis not the zero polynomial, then there are the unique polynomials q(quotient) and r(remainder) such that ( )( )( )( )( )f xr xq xg xg x=+or ( )( )( )( )f xq x g xr x=+where ( )r xeither the zero polynomial or of degree less than the degree of ( )g x. Note: If ( )g xxc=−, then ( )( )()f xq xxcr=−+, where ris a number. If xc=, then ( )f c=Remainder TheoremIf a polynomial ( )f xis divided by ()xc−, then the remainder ( )rf c=. 201 Example: Find the remainder if 43( )62f xxx=−+is divided by (2)x+. Use synthetic division the Remainder Theorem Example: Use the Remainder Theorem to find (3)fif 652( )654161f xxxxx=−+−+
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