1. Section 6.6 The Fundamental Theorem of Algebra
Obj: To use the Fundamental Theorem of Algebra to solve polynomial equations with complex roots

2. Recall: Complex Numbers = Real #s + Imaginary #s
We have solved polynomial equations and found their roots are included in the set of complex numbers.
Recall: Complex Numbers =
Real #s + Imaginary #s
Therefore, our roots have been:
-integers
-rational numbers
-irrational numbers
-imaginary numbers
But, can all polynomial equations be solved using complex numbers?

3. Carl Friedrich Gauss (1777-1855)
-- says roots of every polynomial equation, even those with imaginary coefficients, are complex numbers.
--developed the Fundamental Theorem of Algebra

4. Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n ≥ 1 with complex coefficients, then P(x) = 0 has at least one complex root.
Corollary
Including imaginary roots and multiple roots, as nth degree polynomial equation has exactly n roots
A corollary is a deduction.
Another way of saying this is: you can factor a polynomial of degree n into n linear factors.

5. Imaginary Root Theorem
If the imaginary number a +bi is a root of a polynomial equation with real coefficients, then the conjugate
a- bi is also a root.
Therefore, imaginary roots always _________________________.

6. Using the Fundamental Theorem of Algebra:
Example 1: For the equation x3 + 2x2 – 4x – 6 = 0, find:
a. Number of complex roots:
b. Number of real roots:
c. Possible rational roots:

7. Example 2: For the equation x4 – 3x3 + x2 – x + 3 = 0, find:
a. Number of complex roots:
b. Number of real roots:
c. Possible rational roots:

8. Finding all zeros of a Polynomial Function:
Example 3: Find the number of complex zeros of
f(x) = x5 + 3x4 – x – 3. FIND ALL ZEROS.

9. f(x) =x3 – 2x2 +4x – 8. FIND ALL ZEROS.
Example 4: Find the number of complex zeros of
f(x) =x3 – 2x2 +4x – 8. FIND ALL ZEROS.

10. CLOSURE: Find the degree
How can you determine how many roots there are for a polynomial equation?
Find the degree

11. Assignment:
Pg 337 # 1-8, 9-15 odd, 21,23