What is Factorisation?

Factorisation

Factorisation is the process of writing an algebraic expression as a product of its factors. It is the reverse process of expansion (removing brackets).

Think of it this way: when you expand, you multiply out brackets. When you factorise, you put expressions back into brackets.

Analogy

Factorisation is like the relationship between multiplication and division. Just as $12 \div 3 = 4$ reverses $3 \times 4 = 12$ , factorisation reverses expansion:

Expanding: $3 (a + 2) = 3 a + 6$
Factorising: $3 a + 6 = 3 (a + 2)$

You start with the result and work backwards to find what was multiplied together to produce it!

Note

An expression is fully factorised when none of the remaining algebraic factors inside the brackets can be factorised any further. This is why finding the Highest Common Factor (HCF) is so important.

Factors and Products in Algebra

Product

When two or more terms are multiplied together, the result is called a product.

Factors

The terms that are multiplied together to give a product are called factors.

For example:

The product of $3 x^{2}$ and $- 2 z$ is $- 6 x^{2} z$ .
$3 x^{2}$ and $- 2 z$ are factors of $- 6 x^{2} z$ .

Note

Notice that $x^{2}$ and $z$ are different variables, so they stay separate in the product — they cannot be combined into $x^{3}$ . Always keep different variables distinct when multiplying!

This language applies to both numbers and algebra:

Numerical example	Algebraic example
$4 \times 3 = 12$	$3 \times x = 3 x$
Factors of 12: 1, 2, 3, 4, 6, 12	Factors of $3 x$ : 1, 3, $x$ , $3 x$

Exam Tip

When identifying factors of an algebraic term, consider both the numerical (number) part and the variable (letter) part separately. Then combine them. Note that we focus on positive factors here — you'll encounter negative factors in more advanced work.

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