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.
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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.
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