3.5 Complex Zeros & the Fundamental Theorem of Algebra

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) = x3 + x2 + 81x + 81

4. Find all the zeros and factor completely. • P(x) = x4 - 3x3 + 7x2 + 21x - 26

5. Find all the zeros and factor completely. • P(x) = 3x5 + 54x3 + 243x

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) = x3 – 100x2 + 32x + 50

10. Every polynomial with real coefficients can be factored into the product of linear and irreducible quadratic factors with real coefficients. • Factor f(x) = x4 + 9x2 – 112 into: • Linear and irreducible quadratic factors • Linear factors with complex coefficients