1. Precalculus Lesson 2.5 The Fundamental Theorem of Algebra

2. Find all real zeros of f(x) = x4 - 3x3 + x - 3

3. Write f(x) = x5 + x3 + 2x2 – 12x + 8 as the product of linear factors, and list all the zeros of f.

4. Write f(x) = x4 + 6x3 + 10x2 + 6x + 9 as the product of linear factors, and list all the zeros of f.

5. Upper and Lower Bound Rules Use synthetic division to verify the upper and lower bounds of the real zeros of f(x) = 2x4 – 8x + 3: Upper Bound: x = 3 Lower Bound: x = -4

6. f(x) = 3x4 – 11x3 + 10x - 4 • List all the possible rational zeros. • Use Descartes Rule of Signs to determine the possible number of positive real, negative real, and imaginary zeros. • Find your upper and lower bounds. • Find all of your zeros.

7. Find the fourth-degree polynomial function with real coefficients that has -1, -1, and 3i as zeros Complex zeros occur in conjugate pairs.

8. Find the polynomial function with real coefficients that has -1, 6 + 5i, and 6 – 5i as zeros.

9. Write the polynomial f(x) = x4 – x2 – 20 as: a product of factors that are irreducible over the rationals As the product of linear factors and quadratic factors that are irreducible over the reals In completely factored form

10. Find all the zeros of f(x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f.