1. Fundamental Theorem of Algebra TS: Demonstrating understanding of concepts Warm-Up: T or F: A cubic function has at least one real root. T or F: A polynomial function can have no complex solutions. T or F: A polynomial function could have only one imaginary solution. T or F: A polynomial could have root 2 as its only irrational solution.

2. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system. Linear Factorization Theorem If f(x) is a polynomial of degree n where n > 0, f has precisely n linear factors f(x) = a n (x – c 1 )(x – c 2 )∙∙∙(x – c n ) where c 1, c 2, …, c n are complex numbers.

3. Find a cubic polynomial with zeros of 2i and 3

4. Find the quartic polynomial with zeros -√2 and i, which passes through (1, 6)

5. Factoring Polynomials so they are irreducable over the rationals, reals and complex zeros. Factor each: a) x 4 – x 2 – 20

6. Factoring Polynomials so they are irreducable over the rationals, reals and complex zeros. Factor each: b)x 4 – 3x 3 – x 2 – 12x – 20 (Hint: x 2 + 4 is a factor)

7. You Try: 1) If -1 – 3i is a zero of x 3 + 4x 2 + 14x + 20, find the other zeros 2) Factor the following: x 4 + 6x 2 – 27 a) Irreducible over the rationals: b) Irreducible over the reals: c) Irreducible over the complex: