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Factoring Expressions (Mathematics Lesson)

Factoring Expressions

Factoring is a way of simplifying an expression into 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 (or subtracted) together 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, the a term, which is common to ab and ac, has been taken outside of a bracket. This leaves b + c inside the bracket.

    a is multiplying (b + c). They are factors of a product.

How to Factor an Expression

Fac Question: Factor the expression below.



Step 1
Find the Greatest Common Factor of terms in the expression.
The terms in the expression are 2x2 and 2xy.

The Greatest Common Factor is 2x:

Step 2
Write the Greatest Common Factor found in Step 1 outside of the bracket.



Step 3
Divide each term by the Greatest Common Factor found in Step 1, to see what remains of the term.

1st term

Divide the 1st term in the expression by the Greatest Common Factor.



This becomes the 1st term inside the bracket.



2nd term

Divide the 2nd term in the expression by the Greatest Common Factor.



This becomes the 2nd term inside the bracket.



The expression has been factored.



Note: To check that the expression has been factored correctly, expand the bracket to see if the original expression is obtained.

A Real Example of How to Factor an Expression

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

How to Factor an Expression into Two Brackets

Quadratic equations can be factored into two brackets.



A quadratic equation can be solved using factoring.

Read more about how to solve quadratic equations using factoring

How to Factor a Difference of Squares

The difference of squares is a special type of quadratic equation:



A difference of squares always factors to a simple expression:



Read more about how to factor a difference of squares
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Note
FACTORING & FACTORISING, TO FACTOR & TO FACTORISE

This method has slightly different terms depending on which country you are from.

In American English, the method is factoring and the verb is 'to factor'.

In British English, the method is factorising and the verb is 'to factorise'.

WHAT IS THE GREATEST COMMON FACTOR IN ALGEBRA?

The Greatest Common Factor 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 found in Step 1 must be found for all the terms in the expression.