Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra

1. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Complex Numbers • The complex number system includes real and imaginary numbers. • Standard form of a complex number is: a + bi. • a and b are real numbers. • i is the imaginary unit ( Fundamental Theorem of Algebra Every complex polynomial function of degree 1 or larger (no negative integers as exponents) has at least one complex zero.

2. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Theorem Every complex polynomial function of degree n  1 has exactly n complex zeros, some of which may repeat. Conjugate Pairs Theorem • If is a zero of a polynomial function whose coefficients are real numbers, then the complex conjugate is also a zero of the function. Examples • 1) A polynomial function of degree three has 2 and 3 + i as it zeros. What is the other zero?

3. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Examples • 2) A polynomial function of degree 5 has 4, 2 + 3i, and 5i as it zeros. What are the other zeros? • 3) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What are the zeros?

4. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Examples • 4) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What is the function?

5. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Find the remaining complex zeros of the given polynomial functions 5)

6. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Long Division

7. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Find the complex zeros of the given polynomial functions 6) Possible solutions: Try: Try:

8. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Try:

9. Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Complex zeros: