Math 41 Section 5.5

Math 41 Section 5.5 - October 9 2007 5.5 The Real Zeros of...

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October 9, 2007 5.5 The Real Zeros of a Polynomial Function Use the Remainder Factor Theorems When we divide one polynomial (the dividend) by another (the divisor), we obtain a quotient polynomial and a remainder, the remainder being either the zero polynomial or a polynomial whose degree is less than the degree of the divisor. To check our work, we verify that (Quotient)(Divisor) + Remainder = Dividend This check routine is the basis for a famous theorem called the division algorithm for polynomials , which we state without proof. Division Algorithm for Polynomials If f(x) and g(x) denote polynomial functions and id 6(x) is not a zero polynomial, there are unique polynomial functions q(x) and r(x) such that where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x) f(x) is the dividend g(x) is the divisor q(x) is the quotient Remainder theorem: Let f be a polynomial. Iff (x) is divided be x-c, then the remainder is f(c) Factor theorem: Let f be a polynomial function. Then x-c is a factor of f(x) iff f(c) = 0.
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