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.

3. 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 1 4 4 1 4 4
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 – 9x = 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 odd and odd