1. 6.6 The Fundamental Theorem of Algebra

2. Fundamental Theorem of Algebra
A polynomial of degree > 1 has at least one complex root
Including complex roots and multiple roots, the n degree of the polynomial equation has exactly n roots. The related function has exactly n zeros.

3. Example 1: Using the Fundamental Theorem of Algebra
For the equation x4 – 3x3 + 4x + 1 = 0, find the number of complex roots, the possible number of real roots, and the possible rational roots.
Number of Complex Roots: ________________________
Possible Number of Real Roots: ____________________
List the Possible Rational Roots: ___________________ 4
4, 2, 0 +1
By the Imaginary Root Theorem, the equation has either no imaginary roots, two imaginary roots (one conjugate pair), or four imaginary roots (two conjugate pairs). So the equation has either zero real roots, two real roots, or four real roots.

4. Example 2: Finding All Zeros of a Polynomial Function
y = 2x3 + x2 +1
Step 1: List the possible rational roots.
Step 2: Test each possible rational root until you find a root
Step 3: Use synthetic division with the root you found in Step 2
Step 4: Find the rest of the roots by solving (use quadratic formula if necessary)
Step 1: +1, + ½
Step 2: 2( )3 + ( )2 +1 = ____
2( )3 + ( )2 +1 = ____
Step 3:
___________
Step 4: