PP Section 12.3

# PP Section 12.3 - The Remainder Theorem and the Factor...

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The Remainder Theorem and the Factor Theorem

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Remainder Theorem Let f be a polynomial function. If f ( x ) is divided by x - c , then the remainder is f ( c ). f x x x x ( ) = + - - 3 9 18 24 3 2 Find the remainder if is divided by x + 3 . x + 3 = x - (-3) f ( ) ( ) ( ) ( ) - = - + - - - - = 3 3 3 9 3 18 3 24 30 3 2 24 18 9 3 30 ) 3 )( 18 (3x 2 3 2 - - + = + + - x x x x Check:
Comment Since we want the remainder of dividing by x + 3 , we could also use synthetic division to find it. 3 9 - 18 - 24 - 3 3 -9 0 0 -18 54 30 Thus, the remainder is 30.

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Factor Theorem Let f be a polynomial function. x - c is a factor of f ( x ) if and only if f ( c ) = 0. In other words, if f (c) = 0, then the remainder found if f (x) is divided by x - c is zero. Hence, since x - c divided into f ( x ) evenly (remainder 0), x - c is a factor of f ( x ).
x + 4 = x – (-4) f(-4) = 3(-4) 3 +9(-4) 2 –18(-4) – 24 = 0 Since the remainder is 0, x + 4 is a factor . factor.

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PP Section 12.3 - The Remainder Theorem and the Factor...

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