1. Warm-up 11-3

2. Lesson 11-3 Theorems about Roots
Objectives
Students will:
Find the roots and and multiplicity of roots
Determine the number of roots
Find all roots given some roots and the degree
Find polynomials with specific roots

3. Theorems:
P(x) with degree n > 1 has exactly n factors
► Factor may occur more than once → multiplicity
► (x – 2)2 → 2 occurs twice
If a + bi is a root, then a – bi is a root
If is a root, then is a root

4. Example 1 Find the roots of P(x) = x4 – 2x2 + 1, give any multiplicy
Example 2 P(x) has degree 5 with roots 7, 3 + 2i, and 2 +
Find the other roots

5. General Steps:
Know the number of factors
Look at any given factors
List any roots, or conjugates of imaginary or complex numbers
Use division with given factors to find other factors

6. Example 3 P(x) = x4 – 3x3 – 2x2 + 2x + 12, Find all roots given -1 – i is a root.
HW 107