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# section34 - f then p must divide a and q must divide a n...

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MAT 111 - College Algebra Section 3.4 Zeros of Polynomial Functions An n th degree polynomial can have at most n real zeros. Examples 1. f ( x ) = x 2 - 1 2. g ( x ) = ( x - 1) 2 3. h ( x ) = x 3 - 1 Above statement can be improved: An n th degree polynomial has precisely n zeros if we count the multiplicity of each zero. Any polynomial function with real coeﬃcients can be uniquely factored into a product of linear factors and/or irreducible quadratic factors. Illustrations: (a) f ( x ) = x 2 - 1 = ( x - 1)( x + 1) (b) g ( x ) = x 5 - 5 x 4 + 12 x 3 - 24 x 2 + 32 x - 16 = ( x - 1)( x - 2) 2 ( x 2 + 4) (c) h ( x ) = x 4 + 2 x 2 + 1 = ( x 2 + 1)( x 2 + 1)

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Linear Factorization Theorem: If f ( x ) is a polynomial of degree n > 0, then f has precisely n linear factors f ( x ) = a ( x - c 1 )( x - c 2 )( x - c 3 ) . . . ( x - c n ) where c 1 , c 2 , . . . , c n are complex numbers. Rational Zeros Theorem: If f ( x ) = a n x n + a n - 1 x n - 1 + . . . + a 1 x + a 0 is a polynomial of degree n 1 and if p q , in lowest terms, is a rational zero of
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Unformatted text preview: f , then p must divide a and q must divide a n . Example: List the rational zeros of f ( x ) = x 3 + x 2-4 x-4. Complex Zeros: Consider f ( x ) = x 2 + 4. What are the zeros of f ? If a + bi , where b 6 = 0 , is a zero of a polynomial function f , then . . . is also a zero of the function. Remark: A polynomial function with real coeﬃcients of odd degree has at least one real zero. Examples: 1. Given that 1-√ 3 i is a zero of h ( x ) = 3 x 3-4 x 2 + 8 x + 8, ﬁnd the remaining zeros. 2. Find a polynomial function f with integer coeﬃcients that has the zeros 2, -1, 3 + √ 2 i and whose graph passes through (0, -44)....
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section34 - f then p must divide a and q must divide a n...

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