Section 2-6Section 2-6Finding Complex ZerosFinding Complex Zeros

Section 2-6Section 2-6 Fundamental Theorem of AlgebraFundamental Theorem of Algebra Linear Factorization TheoremLinear Factorization Theorem Complex conjugate zerosComplex conjugate zeros Finding a poly from given zerosFinding a poly from given zeros Factoring a poly with complex zerosFactoring a poly with complex zeros Factoring a poly with real Factoring a poly with real

coefficients coefficients

Fundamental Theorem of Fundamental Theorem of AlgebraAlgebra

a poly function of degree a poly function of degree nn has has nn complex zeroscomplex zeros

some of these zeros may be real and some of these zeros may be real and some may be nonreal (contain i) some may be nonreal (contain i)

the real zeros represent x-the real zeros represent x-intercepts, but the nonreal zeros do intercepts, but the nonreal zeros do not show in the graphnot show in the graph

Fundamental Theorem of Fundamental Theorem of AlgebraAlgebra

the graph of the function shows that the graph of the function shows that there is only one real zero, so the other there is only one real zero, so the other two zeros must be nonrealtwo zeros must be nonreal

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Linear Factorization Linear Factorization TheoremTheorem

if a poly has degree if a poly has degree nn then it has then it has nn linear linear factorsfactors

if zif z11, z, z22, . . . , z, . . . , znn are the complex zeros are the complex zeros and and aa is the leading coefficient then is the leading coefficient then ff ( (xx) ) factors intofactors into

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Complex Conjugate ZerosComplex Conjugate Zeros if a + bi is a zero of if a + bi is a zero of ff ( (xx) then, then a – bi is ) then, then a – bi is

also a zero of also a zero of ff ( (xx) )

Ex. Write a poly in standard form with real Ex. Write a poly in standard form with real coefficients whose zeros include -3, 4, and 2 coefficients whose zeros include -3, 4, and 2

- i- i

Find all the zeros and write the linear factorization

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zeros theof one be toknown is 21 if zeros theall Find6514174)( 24

ixxxxf

Factoring with Real Factoring with Real CoefficientsCoefficients

the linear factorization theorem the linear factorization theorem explains how to factor poly’s, but explains how to factor poly’s, but sometime the factors do not have sometime the factors do not have real coefficientsreal coefficients

it’s also possible to factor a poly it’s also possible to factor a poly using only linear factors and using only linear factors and irreducible quadratic factor, all irreducible quadratic factor, all having real coefficients having real coefficients

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scoeff.' realonly usingFactor 2345 xxxxxxf