Section 2-6Section 2-6
Finding Complex ZerosFinding Complex Zeros
Section 2-6Section 2-6
Fundamental Theorem of Algebra Fundamental Theorem of Algebra
Linear Factorization Theorem Linear Factorization Theorem
Complex conjugate zeros Complex conjugate zeros
Finding a poly from given zeros Finding a poly from given zeros
Factoring a poly with complex zeros Factoring a poly with complex zeros
Factoring a poly with real coefficients Factoring a poly with real coefficients
Fundamental Theorem of AlgebraFundamental Theorem of Algebra
a poly function of degree n has n complexzeros a poly function of degree n has n complexzeros
some of these zeros may be real andsome may be nonreal (contain i) some of these zeros may be real andsome may be nonreal (contain i)
the real zeros represent x-intercepts, butthe nonreal zeros do not show in thegraph the real zeros represent x-intercepts, butthe nonreal zeros do not show in thegraph
Fundamental Theorem of AlgebraFundamental Theorem of Algebra
the graph of the function shows that there isonly one real zero, so the other two zeros mustbe nonreal the graph of the function shows that there isonly one real zero, so the other two zeros mustbe nonreal
Linear Factorization TheoremLinear Factorization Theorem
if a poly has degree n then it has n linearfactors if a poly has degree n then it has n linearfactors
if z1, z2, . . . , zn are the complex zerosand a is the leading coefficient then f (x)factors into if z1, z2, . . . , zn are the complex zerosand a is the leading coefficient then f (x)factors into
Complex Conjugate ZerosComplex Conjugate Zeros
if a + bi is a zero of f (x) then, then a – bi isalso a zero of f (x) if a + bi is a zero of f (x) then, then a – bi isalso a zero of f (x)
Ex. Write a poly in standard form with realcoefficients whose zeros include -3, 4, and 2 - iEx. Write a poly in standard form with realcoefficients whose zeros include -3, 4, and 2 - i
Find all the zeros and write the linear factorization
Factoring with Real CoefficientsFactoring with Real Coefficients
the linear factorization theorem explainshow to factor poly’s, but sometime thefactors do not have real coefficients the linear factorization theorem explainshow to factor poly’s, but sometime thefactors do not have real coefficients
it’s also possible to factor a poly usingonly linear factors and irreduciblequadratic factor, all having realcoefficients it’s also possible to factor a poly usingonly linear factors and irreduciblequadratic factor, all having realcoefficients
