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9-79-7FundamentalFundamentalTheorem ofTheorem of

AlgebraAlgebra

Intro• Find all zeros for each of the following:

MultiplicityMultiplicity – When more than one zero – When more than one zero occurs at the same spotoccurs at the same spot

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Fundamental Theorem of AlgebraFundamental Theorem of AlgebraNumber of Zeros of a Polynomial Theorem:Number of Zeros of a Polynomial Theorem:

A polynomial of degree n A polynomial of degree n >> 1 has exactly n 1 has exactly n complex zeros, if multiplicities are counted.complex zeros, if multiplicities are counted.

TheoremTheorem - A polynomial of degree n has at - A polynomial of degree n has at most n real zerosmost n real zeros

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Multiplicity and DegreeMultiplicity and Degree

What are the zeros of f(x)?What are the zeros of f(x)?x = -1/2; 3; 3/4x = -1/2; 3; 3/4

Which zero has a mulitplicty of 3Which zero has a mulitplicty of 3x = -1/2x = -1/2

What is the degree of f(x)?What is the degree of f(x)?deg = 3 + 2 + 1 = 6deg = 3 + 2 + 1 = 6

How many x-intercepts does f(x) have?How many x-intercepts does f(x) have?33

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Conjugate Zeros TheoremConjugate Zeros TheoremConjugate Zeros Theorem:Conjugate Zeros Theorem:

If a + bIf a + bii is a zero of of p(x) then a – b is a zero of of p(x) then a – bii is is also a zero of p(x)also a zero of p(x)

Application…Application…Find all zeros and their multiplicities of:Find all zeros and their multiplicities of:

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Application….Application….LetLeta)a)Verify that -3Verify that -3ii is a zero is a zero

b)b)Find the remaining zeros of p(x) and their Find the remaining zeros of p(x) and their multiplicities. multiplicities.

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Application• Find all zeros of aaat 25)( 5