1. Section 2-6 Finding Complex Zeros

2. Section 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

3. Fundamental Theorem of Algebra a poly function of degree n has n complex zeros a poly function of degree n has n complex zeros some of these zeros may be real and some may be nonreal (contain i) some of these zeros may be real and some may be nonreal (contain i) the real zeros represent x-intercepts, but the nonreal zeros do not show in the graph the real zeros represent x-intercepts, but the nonreal zeros do not show in the graph

4. Fundamental Theorem of Algebra the graph of the function shows that there is only one real zero, so the other two zeros must be nonreal the graph of the function shows that there is only one real zero, so the other two zeros must be nonreal

5. Linear Factorization Theorem if a poly has degree n then it has n linear factors if a poly has degree n then it has n linear factors if z 1, z 2,..., z n are the complex zeros and a is the leading coefficient then f (x) factors into if z 1, z 2,..., z n are the complex zeros and a is the leading coefficient then f (x) factors into

6. Complex Conjugate Zeros if a + bi is a zero of f (x) then, then a – bi is also a zero of f (x) if a + bi is a zero of f (x) then, then a – bi is also a zero of f (x)

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

8. Find all the zeros and write the linear factorization

11. Factoring with Real Coefficients the linear factorization theorem explains how to factor poly’s, but sometime the factors do not have real coefficients the linear factorization theorem explains how to factor poly’s, but sometime the factors do not have real coefficients it’s also possible to factor a poly using only linear factors and irreducible quadratic factor, all having real coefficients it’s also possible to factor a poly using only linear factors and irreducible quadratic factor, all having real coefficients