1. Copyright © Cengage Learning. All rights reserved. Factoring Polynomials and Solving Equations by Factoring 5

2. Copyright © Cengage Learning. All rights reserved. Section 5.7 Solving Equations by Factoring

3. 3 Objectives 1.Solve a quadratic equation in one variable using the zero-factor property. 2.Solve a higher-order polynomial equation in one variable. 1 1 2 2

4. 4 Solving Equations by Factoring Linear Equations: contain first-degree polynomials 3x + 2 = 0 9x – 6 = 0 Quadratic Equations: contain second-degree polynomials 9x 2 – 6x = 0 3x 2 + 4x – 7 = 0

5. 5 Solving Equations by Factoring Quadratic Equations A quadratic equation in one variable is an equation of the form ax 2 + bx + c = 0(This is called quadratic form.) where a, b, and c are real numbers, and a  0.

6. 6 Solve a quadratic equation in one variable using the zero-factor property 1.

7. 7 Solve a quadratic equation in one variable using the zero-factor property Many quadratic equations can be solved by factoring. For example, to solve the quadratic equation x 2 + 5x – 6 = 0 which is already in quadratic form, we begin by factoring the trinomial and writing the equation as (1) (x + 6)(x – 1) = 0 This equation indicates that the product of two quantities is 0. However, if the product of two quantities is 0, then at least one of those quantities must be 0. This fact is called the zero-factor property.

8. 8 Solve a quadratic equation in one variable using the zero-factor property Zero-Factor Property Suppose a and b represent two real numbers. If ab = 0, then a = 0 or b = 0. By applying the zero-factor property to Equation 1, we have x + 6 = 0 or x – 1 = 0 We can solve each of these linear equations to get x = –6or x = 1

9. 9 Solve a quadratic equation in one variable using the zero-factor property To check, we substitute –6 for x, and then 1 for x in the original equation and simplify. For x = –6 For x = 1 x 2 + 5x – 6 = 0 x 2 + 5x – 6 = 0 (–6) 2 + 5(–6) – 6 ≟ 0 (1) 2 + 5(1) – 6 ≟ 0 36 – 30 – 6 ≟ 0 1 + 5 – 6 ≟ 0 6 – 6 ≟ 0 6 – 6 ≟ 0 0 = 0 0 = 0 Both solutions check.

10. 10 Solve a quadratic equation in one variable using the zero-factor property The quadratic equations 9x 2 – 6x = 0 and 4x 2 – 25 = 0 are each missing a term. The first equation is missing the constant term, and the second equation is missing the term involving x. These types of equations often can be solved by factoring.

11. 11 Example Solve: 9x 2 – 6x = 0 Solution: We begin by factoring the left side of the equation. 9x 2 – 6x = 0 3x(3x – 2) = 0 By the zero-factor property, we have 3x = 0 or 3x – 2 = 0 Factor out the common factor of 3x.

12. 12 Example 1 – Solution Solve each linear equation. 3x = 0 or 3x – 2 = 0 Add 2 to both sides. cont’d Divide both sides of each equation by 3.

13. 13 Example 1 – Solution Check: We substitute these results for x in the original equation and simplify. Both solutions check. cont’d

14. 14 Solve a quadratic equation in one variable using the zero-factor property Some equations must be simplified before we write them in quadratic form. Many times this requires the distributive property.

15. 15 Solve a higher-order polynomial equation in one variable 2.

16. 16 Solve a higher-order polynomial equation in one variable A higher-order polynomial equation is any equation in one variable with a degree of 3 or larger.

17. 17 Example 6 Solve: x 3 – 2x 2 – 63x = 0 Solution: We begin by completely factoring the left side. x 3 – 2x 2 – 63x = 0 x(x 2 – 2x – 63) = 0 x(x + 7)(x – 9) = 0 x = 0or x + 7 = 0 or x – 9 = 0 x = –7 x = 9 Check each solution. Factor out x, the GCF. Factor the trinomial. Set each factor equal to 0. Solve each linear equation.

18. 18 Solve a higher-order polynomial equation in one variable As with quadratic equations, higher-order equations in one variable must be set equal to zero to solve by factoring.