Remainder and Factor Theorem

From Department of Mathematics at UTSA
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The polynomial division algorithm is as follows: suppose and are nonzero polynomials where the degree of is greater than or equal to the degree of . Then there exist two unique polynomials, and , such that , where either or the degree of is strictly less than the degree of .

Remainder Theorem: Suppose is a polynomial of degree at least 1 and c is a real number. When is divided by Failed to parse (syntax error): {\displaystyle x − c } the remainder is .

Proof: By the division algorithm, , where r must be a constant since has a degree of 1. must hold for all values of , so we can set and get that . Thus the remainder .

Factor Theorem: Suppose is a nonzero polynomial. The real number is a zero of if and only if Failed to parse (syntax error): {\displaystyle (x − c) } is a factor of .

By the division algorithm, is a factor of if and only if . So, since when is divided by , is a factor of if and only if ; that is, if is a zero of .

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