1. Objective: Students will be able to use the rational root theorem and the irrational root theorem to solve polynomial equations, and can identify the multiplicity of roots.

2. Part 1 – Finding Zeros of Polynomial Equations Remember that you can find the zeros (roots) of any equation by setting each factor equal to 0 and solving for x. Directions: Find the roots of each polynomial equation by factoring. Example 1: 3x 5 + 18x 4 + 27x 3 = 0

3. Example 2 x 3 – 2x 2 – 25x = -50

4. Example 3 2x 3 – 12x 2 = 32x – 192

5. Multiplicity of roots Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. Look back at example 1, do you see a multiple root? Multiplicity – the multiplicity of root r is the number of times that x – r is a factor of P(x).

6. Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. Example 4: x 3 – 9x 2 + 27x – 27 = 0

7. Example 5 x 3 + 6x 2 + 12x + 8 = 0

8. Example 6 5x 4 – 20x 3 + 20x 2 = 0

9. Homework for tonight Homework # ______ Textbook pg. 186 #2, 3, 15, 16, 17

10. Rational Root Theorem If the polynomial you are given is not factorable, the Rational Root Theorem can help you find all possible rational roots of a polynomial equation. Example 7: A popcorn producer is designing a new box for the popcorn. The marketing department has designed a box with the width 2 inches less than the length and with the height 5 inches greater than the length. The volume of each box must be 24 cubic inches. What is the length of the box?

11. Example 8 The design of a box specifies that its length is 4 inches greater than its width. The height is 1 inch less than the width. The volume of the box is 12 cubic inches. What is the width of the box?

12. You try… A box is 2 inches longer than its height. The width is 2 inches less than the height. The volume of the box is 15 cubic inches. How tall is the box?