1. Objectives Identify the multiplicity of roots.
Use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. 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). 

8. 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). 

9. Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation.

10. Polynomial equations may also have irrational roots.

11. Example 4: Identifying All of the Real Roots of a Polynomial Equation
Identify all the real roots of 2x3 – 9x2 + 2 = 0.
Step 1 Use the Rational Root Theorem to identify possible rational roots.
±1, ±2
= ±1, ±2, ± . 1 2
p = 2 and q = 2
Step 2 Graph y = 2x3 – 9x2 + 2 to find the x-intercepts.
The x-intercepts are located at or near –0.45, 0.5, and The x-intercepts –0.45 and 4.45 do not correspond to any of the possible rational roots.

12. Step 3 Test the possible rational root .
Example 4 Continued 1 2
Step 3 Test the possible rational root . 1 2
Test The remainder is 0, so (x – ) is a factor. 1 2
2 – 1 –4 –2 2 –8 –4
The polynomial factors into (x – )(2x2 – 8x – 4). 1 2
Step 4 Solve 2x2 – 8x – 4 = 0 to find the remaining roots.
2(x2 – 4x – 2) = 0
Factor out the GCF, 2
Use the quadratic formula to identify the irrational roots. 4± 2 x
= = ±

13. ( ) Example 4 Continued The fully factored equation is 1
2 x – x – x – 2 – = 0 2 æ é ç ë è ö ÷ ø ù û
The roots are , , and 1 2 + -

14. 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
4. 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?
3 in.
5. Identify all the real roots of x3 + 5x2 – 3x – 3 = 0.
1, ,