Factoring an Expression
(KS3, Year 7)
Factoring is a way of simplifying an expression. Factoring writes an expression as a product of factors. Imagine we wanted to factor the expression below. We would take out a common factor from each term outside of a bracket, leaving another expression inside the bracket. ab plus ac equals a (b plus c) It is the opposite of expanding brackets.

Understanding Factoring an Expression

Factoring allows us to change an expression where terms are added together (or subtracted from each other) to one where they are multiplied together. Addition of ab and bc becomes the multiplication of a and (b plus c)
  • On the left-hand side of the equation, ab and ac are added together.
  • On the right-hand side of the equation, a is multiplying (b + c).
Factoring has been achieved by taking the a, which is common to ab and ac, outside of the bracket. This leaves b + c inside the bracket. a is the Greatest Common Factor of ab and ac. It is the largest factor that is common to the two terms.

How to Factor an Expression

Factoring an expression is easy.

Question

Factor the expression below.
factor an expression example

Step-by-Step:

1

Find the Greatest Common Factor of the terms in the expression. The terms in the expression are 2x2 and 2xy.
  • Look at the letters that appear in both terms. In our example, only x appears in both terms. For each letter that appears in both terms, find the letter with the smallest exponent.
    2 x2 , 2 x y
    x has the smallest exponent. (x has an implicit exponent of 1 (x = x1), whereas x2 has an exponent of 2). factor_an_expression_step_1_x

2

Write the Greatest Common Factor outside of the brackets. factor_an_expression_step_2

3

Divide each term in the original expression by the Greatest Common Factor.

1st Term

Divide the 1st term (2x2) in the original expression by the Greatest Common Factor (2x). factor an expression step 3 1st bracket 1 Write this as the 1st term inside the bracket. factor an expression step 3 1st bracket 2

2nd Term

Divide the 2nd term (2xy) in the original expression by the Greatest Common Factor (2x). factor an expression step 3 2nd bracket 1 Write this as the 2nd term inside the bracket, and add to the previous result. factor an expression step 3 2nd bracket 2 Note: The term is added because there is a + before the second term in the original expression.

Answer:

We have factored the expression.

factor an expression answer Check that the expression has been factored correctly by expanding the brackets to see if you get the original expression.

Lesson Slides

The slider below shows another real example of how to factor an expression.

Factoring, Factorising

The method is refered to as 'factoring' or 'factorising'. The verb is 'to factor' or 'to factorise'.

What Is the Greatest Common Factor in Algebra?

The greatest common factor in algebra is the largest factor that is common to two or more terms.

Expressions with More than Two Terms

Expressions can have more than two terms that are added or subtracted together. These expression are factored in the same way. The Greatest Common Factor must be found for all the terms in the expression.
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This page was written by Stephen Clarke.