ss_3_4

# ss_3_4 - Section 3.4 The Real Zeros of a Polynomial...

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Example. If f ( x )= x 3 +2 x 2 + x - 1and g ( x )= x 2 +1,then f ( x ) g ( x ) = x +2 - 3 x 2 +1 = q ( x )+ r ( x ) g ( x ) where q ( x )= x + 2 (Quotient) r ( x )= - 3 (Remainder) In general if f ( x )and g ( x ) are polynomials, then f ( x ) g ( x ) = q ( x )+ r ( x ) g ( x ) where r ( x ) is a polynomial of degree less than the degree of g ( x ). If g ( x )= x - c in the above, then f ( x ) x - c = q ( x )+ r ( x ) x - c and so f ( x )= q ( x )( x - c )+ r ( x ) Setting x = c in the last expression above gives f ( c )= r ( c ). The above proves the following theorem. The Remainder Theorem. If f ( x ) is a polynomial, then the remainder of f ( x ) x - c is f ( c ). The following examples illustrate the Remainder Theorem. Example. x 3 - 2 x 2 +3 x - 1 x - 3 has remainder of 17 because x 3 - 2 x 2 +3 x - 1=17when x =3. Example. x 3 - 2 x 2 +3 x - 1 x +3 has remainder of -55 because x 3 - 2 x 2 +3 x - 1= - 55 when x = - 3. The Remainder Theorem implies the following theorem. The Factor Theorem. The term ( x - c ) is a factor of a polynomial f ( x ) iﬀ (iﬀ means if and only if) f ( c )=0. The following examples illustrate the Factor Theorem. Example. x 3 - 2 x 2 +3 x - 1 6 =0when x = 3. Hence ( x - 3) is not a factor of x 3 - 2 x 2 +3 x - 1. Example. x 3 - 2 x 2 +3 x - 1 6 =0when x = - 3. Hence ( x + 3) is not a factor of x 3 - 2 x 2 +3 x - 1. Example. x 2 - 9=0when x = 3. Hence ( x - 3) is a factor of x 2 - 9. Example.

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## This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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ss_3_4 - Section 3.4 The Real Zeros of a Polynomial...

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