7.4 Solving Polynomial Equations

1. 7.4 Solving Polynomial Equations Objectives: Solve polynomial equations Find the real zeros of polynomial functions and state the multiplicity of each

2. Example 1 Use factoring to solve 5x3 – 12x2 + 4x = 0. x(5x2 – 12x + 4) = 0 x(5x – 2)(x – 2) = 0 x = 0 , x = 2

3. 1 Example 2 Use a graph, synthetic division, and factoring to find all roots of x3 + 3x2 – 4 = 0. First, graph the polynomial function to approximate the roots. Then use synthetic division to test your choices. 1 3 0 -4 1 4 4 1 4 4 0 Since the remainder is 0, x – 1 is a factor of x3 + 3x2 - 4.

4. Example 2 Use a graph, synthetic division, and factoring to find all roots of x3 + 3x2 – 4 = 0. Since the remainder is 0, x – 1 is a factor of x3 + 3x2 - 4. x3 + 3x2 – 4 = 0 (x – 1)(x2 + 4x + 4) = 0 (x – 1)(x + 2)(x + 2) = 0 x = 1 x = -2 x = -2 The roots of x3 + 3x2 – 4 are 1 and -2, with the root -2 occurring twice.

5. Practice Use a graph, synthetic division, and factoring to find all of the roots of x3 + 2x2 – 4x – 8 = 0.

6. Example 3 Use variable substitution and factoring to find all roots of x4 – 5x2 – 6 = 0. Substitute u for x2 in the above equation and then solve for u. (x2)2 – 5(x2) – 6 = 0 u2 – 5u – 6 = 0 (u – 6)(u + 1) = 0 u = 6 u = -1 x2 = 6 x2 = -1

7. Practice Use variable substitution and factoring to find all of the roots of x4 – 9x2 + 14 = 0.

8. Example 4 Find the real zeros of the function. Give approximate values to the nearest hundredth, if necessary. f(x) = x4 – 10x3 + 22x2 + 20x - 48 x = 4 x = 6 x = 1.41 x = -1.41

9. Homework Lesson 7.4 Exercises 11-25 odd and 39-49 odd