1. Finding Real Roots of Polynomial Equations 6-5
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up Factor completely. 1. 2y3 + 4y2 – 30y 2y(y – 3)(y + 5)
2. 3x4 – 6x2 – 24
3(x – 2)(x + 2)(x2 + 2)
Solve each equation.
3. x2 – 9 = 0
x = – 3, 3
4. x3 + 3x2 – 4x = 0
x = –4, 0, 1

3. Objectives Identify the multiplicity of roots.
Factor polynomial expressions and find the roots.

4. Vocabulary multiplicity
The number of times that x-r is a factor of P(x)
When a real root has even multiplicity, the graph of P(x) touches the x-axis but does not cross it.
When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis.

5. In Lesson 6-4, you used several methods for factoring polynomials
In Lesson 6-4, you used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots.
Recall the Zero Product Property from Lesson 5-3. You can find the roots, or solutions, of the polynomial equation P(x) = 0 by setting each factor equal to 0 and solving for x.
ROXS

6. Example 1A: Using Factoring to Solve Polynomial Equations
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
4x4(x2 + x – 6) = 0
Factor out the GCF, 4x4.
4x4(x + 3)(x – 2) = 0
Factor the quadratic.
4x4 = 0 or (x + 3) = 0 or (x – 2) = 0
Set each factor equal to 0.
x = 0, x = –3, x = 2
Solve for x.
The roots are 0, –3, and 2.

7. Example 1A Continued
Check Use a graph. The roots appear to be located at x = 0, x = –3, and x = 2. 

8. Example 1B: Using Factoring to Solve Polynomial Equations
Solve the polynomial equation by factoring.
x = 26x2
x4 – 26 x = 0
Set the equation equal to 0.
Factor the trinomial in quadratic form.
(x2 – 25)(x2 – 1) = 0
(x – 5)(x + 5)(x – 1)(x + 1)
Factor the difference of two squares.
x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0
x = 5, x = –5, x = 1 or x = –1
Solve for x.
The roots are 5, –5, 1, and –1.

9. Check It Out! Example 1a
Solve the polynomial equation by factoring.
2x6 – 10x5 – 12x4 = 0
2x4(x2 – 5x – 6) = 0
Factor out the GCF, 2x4.
2x4(x – 6)(x + 1) = 0
Factor the quadratic.
2x4 = 0 or (x – 6) = 0 or (x + 1) = 0
Set each factor equal to 0.
x = 0, x = 6, x = –1
Solve for x.
The roots are 0, 6, and –1.

10. Check It Out! Example 1b
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
x3 – 2x2 – 25x + 50 = 0
Set the equation equal to 0.
(x + 5)(x – 2)(x – 5) = 0
Factor.
x + 5 = 0, x – 2 = 0, or x – 5 = 0
x = –5, x = 2, or x = 5
Solve for x.
The roots are –5, 2, and 5.

11. Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3. For example, the root 0 is a factor three times because 3x3 = 0.
The multiplicity of root r is the number of times that x – r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis.

12. You cannot always determine the multiplicity of a root from a graph
You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form.

13. Example 2A: Identifying Multiplicity
Identify the roots of each equation. State the multiplicity of each root.
x3 + 6x2 + 12x + 8 = 0
x3 + 6x2 + 12x + 8 = (x + 2)(x + 2)(x + 2)
x + 2 is a factor three times. The root –2 has a multiplicity of 3.
Check Use a graph. A calculator graph shows a bend near (–2, 0). 

14. Example 2B: Identifying Multiplicity
Identify the roots of each equation. State the multiplicity of each root.
x4 + 8x3 + 18x2 – 27 = 0
x4 + 8x3 + 18x2 – 27 = (x – 1)(x + 3)(x + 3)(x + 3)
x – 1 is a factor once, and x + 3 is a factor three times. The root 1 has a multiplicity of 1. The root –3 has a multiplicity of 3.
Check Use a graph. A calculator graph shows a bend near (–3, 0) and crosses at (1, 0). 

15. x – 2 is a factor four times. The root 2 has a multiplicity of 4.
Check It Out! Example 2a
Identify the roots of each equation. State the multiplicity of each root.
x4 – 8x3 + 24x2 – 32x + 16 = 0
x4 – 8x3 + 24x2 – 32x + 16 = (x – 2)(x – 2)(x – 2)(x – 2)
x – 2 is a factor four times. The root 2 has a multiplicity of 4.
Check Use a graph. A calculator graph shows a bend near (2, 0). 

16. Check It Out! Example 2b
Identify the roots of each equation. State the multiplicity of each root.
2x6 – 22x5 + 48x4 + 72x3 = 0
2x6 – 22x5 + 48x4 + 72x3 = 2x3(x + 1)(x – 6)(x – 6)
x is a factor three times, x + 1 is a factor once, and x – 6 is a factor two times.
The root 0 has a multiplicity of 3. The root –1 has a multiplicity of 1. The root 6 has a multiplicity of 2.

17. Lesson Quiz
Solve by factoring.
1. x3 + 9 = x2 + 9x
–3, 3, 1
Identify the roots of each equation. State the multiplicity of each root.
0 and 2 each with multiplicity 2
2. 5x4 – 20x3 + 20x2 = 0
3. x3 – 12x2 + 48x – 64 = 0
4 with multiplicity 3

18. SUGGESTED PRACTICE
PAGE 442 #1-9, AND 43