1. 3.5 Complex Zeros & the Fundamental Theorem of Algebra

2. Fundamental Theorem of Algebra An nth degree polynomial has exactly n zeros in the complex number system.

3. Find all the zeros and factor completely. P(x) = x 3 + x 2 + 81x + 81 Final Possibilities Total Zeros

4. Find all the zeros and factor completely. P(x) = x 4 - 3x 3 + 7x 2 + 21x - 26 Final Possibilities Total Zeros

5. Find all the zeros and factor completely. P(x) = 3x 5 + 54x 3 + 243x Total Zeros

6. Find the polynomial with zeros at i, -i, 2, -2 and P(5)=273

7. Complex Conjugate Theorem If a + bi is a root then a – bi is also a root.

8. Find the polynomial with zeros at:

9. Use Descartes’ Rule to count the number of real and imaginary zeros. P(x) = x 3 – 100x 2 + 32x + 50 Two changes in sign: 2 or 0 positive real zeros One change in sign: 1 negative real zeros Positive Real ZerosNegative Real ZerosImaginary Zeros

10. Every polynomial with real coefficients can be factored into the product of linear and irreducible quadratic factors with real coefficients. Factor f(x) = x 4 + 9x 2 – 112 into: Linear and irreducible quadratic factors Linear factors with complex coefficients Linear and Quadratic Factors Theorem

11. pg 298 #9, 11, 23-45 odd