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Introduction to Factoring Polynomials

Factoring is the opposite of multiplying, or expanding, an expression. Factors are multiplied together to get a product, so when we factor, we want to split a product into its factors. In this section, we will show you how to factor polynomials using the greatest common factor.

Factoring Algebraic Expressions

Factoring Algebraic Expressions

Examples

Click on the titles below to view each example.

Example 1. Factor the expression 4x cubed plus 20 x squared.

Line 1: Find the greatest common factor of 4x cubed and 20x squared. First, factor each numerical coefficient into primes and write the variables with exponents in expanded form, so the term 4 x cubed is written as 2 times 2 times x times x times x, and the term 20 x squared is written as 2 times 2 times 5 times x times x.

Line 2: Identify the common factors in each term and multiply them together to get the greatest common factor, so the greatest common factor is 2 times 2 times x times x equals 4x squared.

Line 3: Rewrite each term in the expression as a product using the greatest common factor, so the expression is 4x squared times x plus 4 x squared times 5.

Line 4: Use the distributive property in reverse to factor the expression to 4x squared times open parenthesis x plus 5 close parenthesis.

Line 5: Check your answer by expanding your factored expression. To expand, multiply 4x squared by x and by 5, so the expression is 4x cubed plus 20 x squared. Since this is the same expression we started with, we know we have factored correctly.

Example 1. Factor the expression 14 x cubed y squared plus 8 x squared y squared minus 10 x squared y.

Line 1: Find the greatest common factor of 14 x cubed y squared, 8 x squared y squared, and 10 x squared y. First, factor each numerical coefficient into primes and write the variables with exponents in expanded form, so the term 14 x cubed y squared is written as 2 times 7 times x times x times x times y times y, the term 8 x squared y squared is written as 2 times 2 times 2 times x times x times y times y, and the term 10 x squared y is written as 2 times 5 times x times x times y.

Line 2: Identify the common factors in each term and multiply them together to get the greatest common factor, so the greatest common factor is 2 times x times x times y equals 2 x squared y.

Line 3: Rewrite each term in the expression as a product using the greatest common factor, so the expression is 2 x squared y times 7 x y plus 2 x squared y times 4y minus 2 x squared y times 5.

Line 4: Use the distributive property in reverse to factor the expression to 2 x squared y times open parenthesis 7 x y plus 4y minus 5 close parentheses.

Line 5: Check your answer by expanding your factored expression. To expand multiply 2 x squared y by 7 x y, 4y and negative 5, so the expression becomes 14 x cubed y squared plus 8 x squared y squared minus 10 x squared y. Since this is the same expression we started with, we know we have factored correctly.

Activity

Try this activity to test your skills. If you have trouble, check out the information in the module for help.

Summary and Worksheet

Attribution

Examples Source: "Prealgebra - opens in a new window" by Lynn Marecek & Mary Anne Anthony-Smith is licensed under CC BY 4.0 - opens in a new window / A derivative from the original work - opens in a new window