1. Factoring Polynomials
Section 5.3
Factoring Polynomials

2. Objectives Common Factors Factoring and Equations
Factoring by Grouping

3. Common Factors
When factoring a polynomial, we first look for factors that are common to each term. By applying the distributive property, we can write a polynomial as two factors.
For example:
It can be factored as follows:

4. Example
Factor. a. b. c. d. Solution a. b.

5. Example (cont)
Factor. a. b. c. d. Solution c. d.

6. Example
Factor. a. b. Solution a. b.

7. Factoring and Equations
To solve equations using factoring, we use the zero-product property. It states that, if the product of two numbers is 0, then at least one of the numbers must equal 0.

8. Example
Solve each equation. a. b. Solution

9. Example Solve each polynomial equation. a. b. Solution
a. We begin by factoring out the greatest common factor.
b. We begin by factoring out the greatest common factor.
No real number can satisfy x2 = –1, the only solution is 0.

10. Polynomial equations can also be solved numerically and graphically.

12. Example
Solve the equation 6x – x2 = 0 numerically, graphically, and symbolically.
Solution
Numerical: Make a table of values. x y 1 7 1 5 2 8 3 9 4 6
Graphical: Plot the points in the table.
The intercepts are the solution to the equation.

13. Example (cont)
Solve the equation 6x – x2 = 0 numerically, graphically, and symbolically. Solution Symbolic: Start by factoring the left side of the equation.
Note that the numerical and graphical solutions agree with the symbolic solutions.

15. Example
Factor. a. 3x(x + 1) + 4(x + 1) b. 3x2(2x – 1) – x(2x – 1) Solution a. Both terms in the expression contain the binomial x + 1. Use the distributive property to factor. b.

17. Example
Factor the polynomial. Solution

18. Example
Factor the polynomial. Solution