1. Warm-up Multiply the factors and write in standard form.

2. 1. How many zeros does this function have? I. Complex Zeros

3. 2. How many zeros does this function have? Complex Zeros

4. 3. How many zeros does this function have? An example with no real zeros How can you be sure there are no real zeros when you graph it on the calculator?

5. Section 3.7 Complex Zeros 1.Complex Zeros 2.Factoring with Complex Zeros 3.Conjugate Pairs Theorem 4.Finding Complex Zeros 1.Complex Zeros 2.Factoring with Complex Zeros 3.Conjugate Pairs Theorem 4.Finding Complex Zeros

6. If f (x) is a complex polynomial of degree n > 0, f has exactly n linear factors and f has n zeros in the complex plane I. Complex Zeros Fundamental Theorem of Algebra: Every complex polynomial function has at least one complex zero.

7. II. Factoring with complex zeros Find the zeros and write as a product of linear factors.

8. Find rational zeros first. Find all real and complex zeros and factor zeros are at: 4 and -7

9. III. Conjugate Pairs Theorem Complex zeros always come in conjugate pairs! If a + bi is a zero of f, then a – bi is also a zero of f If a + bi is a zero of f, then a – bi is also a zero of f Determine the zeros

10. III. a) Corollary to Conjugate Pairs Theorem Suppose a function has degree and zeros as given, what are the remaining zeros? 1.Degree 3; zeros: 1, 2 + i 2.Degree 6; zeros 3 + 2i, i, -4 + i Can a polynomial of degree 3 have as zeros 2i, 4-i ? A polynomial f of odd degree with real coefficients has at least one real zero. Will a polynomial of degree 4 have real zeros if it has complex zeros 4-i, and 5i ?

11. IV. Linear Factors of Complex Zeros Form a polynomial with real coefficients satisfying : 1)degree 4 and zeros at: 3, multiplicity 2; and 2) degree 3 and zeros at: 4, and

12. V. Given a complex zero, find remaining Complex Zeros Suppose the function has as a zero. Determine the remaining zeros of the function. a)Determine the matching pair of complex zeros. b)Multiply linear factors of the complex zeros. c)Polynomial division. d)Solve for zeros of q(x) a)Determine the matching pair of complex zeros. b)Multiply linear factors of the complex zeros. c)Polynomial division. d)Solve for zeros of q(x)

13. VI. Find Complex Zeros and write in factored form Suppose the function has as a zero. Write in factored form:

14. VI. Find Complex Zeros and write in factored form (p. 237 #29). Suppose the function has as a zero. Determine the remaining zeros of the function Write in factored form