Factoring using a common factor that is an expression

When the common factor is an expression in brackets

Sometimes, the common factor is not just a number or a letter, but an entire expression in brackets, such as (x+2) or (2x-1).

Example: factorise A = (x+2) × 5 + (x+2) × 3x. Here, (x+2) appears in both terms: this is the common factor. A = (x+2)(5+3x)

Method

1. Check that the same expression in brackets appears in each term
2. Move this expression out as a factor in front of one set of brackets
3. In the remaining set of brackets, write what was multiplying each expression

Example with a minus sign

Factorise B = (2x-1) × 7 - (2x-1) × x. B = (2x-1)(7-x)

Cases where you need to pay attention to the sign

Factorise C = (x+3) × x - (x+3). Here, the second term is simply (x+3), which can be written as (x+3)*1. C = (x+3)*x - (x+3)*1 = (x+3)(x-1)

Common pitfall

A common mistake is to forget the implicit ‘*1’ when a term is simply the expression in brackets without a visible coefficient. You must also ensure you use the correct + and – signs in front of each term when factoring: a mistake with the sign completely changes the result. Remember to always check your answer by working it out step by step.