From distributivity to factorisation

Reminder: expanding using the distributive property

You’re already familiar with the distributive property: to expand an expression such as k(a+b), we calculate k(a+b) = ka + kb. For example: 3(x+5) = 3x + 35 = 3x+15.

Factoring: the inverse operation

Factoring an expression means doing the opposite of expanding it. We start with a sum (or a difference) of terms and write it in the form of a product of factors.

Example:

• Expanding: 3(x + 5) = 3x + 15
• Factoring: 3x + 15 = 3(x + 5)

In both cases, the two forms represent the same quantity, but in a different form.

Vocabulary

Term	Meaning
Sum	Expression with + or – between the terms, e.g. 3x + 15
Product	Expression in which one factor multiplies another, e.g. 3(x + 5)
Common factor	A number or letter present in every term of the sum

Why factorise?

Factorising helps to simplify calculations, solve equations, and highlight common factors that will be useful later on (at sixth form level, for solving product equations).

Common pitfall

Do not confuse expanding (product → sum) with factoring (sum → product): these are two opposite operations. A good way to check a factorisation is to expand the result obtained: you should get the original expression back.