The fundamental theorem of algebra states that any polynomial of degree greater than 1 with complex coefficients has at least one complex zero. A related theorem is that a polynomial of degree n has at most n zeros. Another theorem is that a polynomial of degree greater than 1 with complex coefficients has exactly n complex zeros counting multiplicities.

1. The Fundamental Theorem of Algebra If p ( x ) is any polynomial of degree n>1 with complex coefficients, then p ( x ) has at least one complex zero. A related theorem: A polynomial of degree n has at most n zeros.

2. Finding Zeros:

3. Multiplicity of Zeros: If a polynomial has the same zero more than once, we describe that zero’s multiplicity : Ex: Another theorem: A polynomial of degree n >1 with complex coefficients has exactly n complex zeros, if multiplicities are counted.

4. Complex Zeros

8. Factoring Sums and Differences of Powers Recall that can be factored into . What about ?

9. Find the cube roots of 8.