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Dividing Polynomials Section 2.4

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Objectives Divide two polynomials using either long division or synthetic division. Use the Factor Theorem to show that x-c is a factor of a polynomial. Use the Remainder Theorem to evaluate a polynomial at a given value.

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Vocabulary quotient remainder dividend divisor

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Division Algorithm If f (x) and d (x) are polynomials, with d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f (x), then there exists unique polynomials q(x) and r(x) such that f(x) = d(x) · q(x) + r(x) The remainder, r(x), equals 0 or it is of degree less than the degree of d(x). If r(x) = 0, we say that d(x) divides evenly into f(x) and that d(x) and q(x) are factors of f(x).

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Find the quotient and remainder using long division.

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Find the quotient and remainder using synthetic division.

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Remainder Theorem If the polynomial f (x) is divided by x – c, then the remainder is f (c).

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Use division and the Remainder Theorem to evaluate P(c), where

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Factor Theorem Let f (x) be a polynomial a.If f(c) = 0, then x – c is a factor of f (x). b.If x – c, is a factor of f(x), then f(c) = 0.

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Use synthetic division to show that x = 6 is a root of

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