Warm-up 11-3.

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Presentation transcript:

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 is a root, then 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 + 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