# 3_4 - Section 3.4: The Real Zeros of a Polynomial Function...

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Section 3.4: The Real Zeros of a Polynomial Function Instructor: Ms. Hoa Nguyen (nguyen@scs.fsu.edu) 1 The Remainder If f ( x ) and g ( x ) are polynomials, then f ( x ) g ( x ) = q ( x ) + r ( x ) g ( x ) (1) f ( x ): the dividend g ( x ): the divisor ( g ( x ) 6 = 0) q ( x ): the quotient r ( x ): the remainder Remark : r ( x ) has a degree less than the degree of g ( x ). Example 1 Example 2 1

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Example 3 : Find the quotient and the remainder of f ( x ) g ( x ) with f ( x ) = x 3 + 2 x 2 + x - 1 and g ( x ) = x - 2. Compute f (2) then compare with the remainder . The Remainder Theorem: If g ( x ) = x - c then the remainder of f ( x ) g ( x ) is f ( c ). 2
2 The Factor Theorem r is a zero of the polynomial f ( x ) ⇐⇒ f ( r ) = 0 ⇐⇒ the term ( x - r ) is a factor of f . Example 4 Theorem : Every polynomial of real coeﬃcients a n , a n - 1 , ··· , a 1 , a 0 can be factored into factors that are either LINEAR factors or IRREDUCIBLE QUADRATIC factors. An irreducible quadratic factor is a quadratic function whose discriminant (

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## This note was uploaded on 05/23/2011 for the course MAC 1147 taught by Professor Nuegyen during the Spring '11 term at FSU.

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3_4 - Section 3.4: The Real Zeros of a Polynomial Function...

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