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# Math: Basic Tutorials : Introduction to Factoring Polynomials

## 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 Video transcript - RTF

## 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.