Transcript Lesson 11-3 Theorems about Roots

Warm-up 11-3
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
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
a  c b is a root, then
a  c b is a root
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 + 5
Find the other roots
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
Example 3 P(x) = x4 – 3x3 – 2x2 + 2x + 12, Find all
roots given -1 – i is a root.
HW 107