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

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

3. Section 3.7 Complex Zeros
Complex Zeros
Factoring with Complex Zeros
Conjugate Pairs Theorem
Finding Complex Zeros

4. I. Complex Zeros If f (x) is a complex polynomial of degree n > 0,
Fundamental Theorem of Algebra: Every complex polynomial function has at least one complex zero.
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

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

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

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

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

9. IV. Linear Factors of Complex Zeros
Form a polynomial with real coefficients satisfying :
zeros at: 4, and and satisfying f(0)=3
A function with zeros at 2i and -2i will have linear factors (x-2)

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

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

12. 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