Factoring Polynomials Section 5.3 Factoring Polynomials

Objectives Common Factors Factoring and Equations Factoring by Grouping

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:

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

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

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

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.

Example Solve each equation. a. b. Solution

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.

Polynomial equations can also be solved numerically and graphically.

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.

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.

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.

Example Factor the polynomial. Solution

Example Factor the polynomial. Solution