What is Factorisation?

Factorisation

Factorisation is the process of writing an algebraic expression as a product of two or more factors — essentially inserting brackets so that the result is a multiplication, rather than a sum or difference.

Think of factorisation and expansion as opposite operations:

Expansion → remove brackets → $5 x (x + 2 y) = 5 x^{2} + 10 x y$
Factorisation → insert brackets → $5 x^{2} + 10 x y = 5 x (x + 2 y)$

Analogy

Factorisation is like unpacking a lunchbox. Expansion tips everything out onto the table (removes the brackets). Factorisation packs everything neatly back in (puts the brackets back), grouping items that belong together.

An expression is called fully factorised when none of its factors can be broken down any further.

In this topic, we'll cover two main approaches:

HCF factorisation — taking out the highest common factor
Factorisation using algebraic identities — recognising special patterns like the difference of two squares and perfect square trinomials

Key Vocabulary Review

Before diving in, let's make sure the key vocabulary is solid:

Factor

A factor is a number or expression that divides exactly into another. For example, 5 and $x$ are both factors of $5 x$ .

HCF (Highest Common Factor)

The HCF of two or more terms is the largest factor that divides into all of them exactly. It can include both numbers and variables.

Coefficient

The numerical part of an algebraic term. In $7 x^{2}$ , the coefficient is 7.

Algebraic Identity

An algebraic identity is an equation that is always true, no matter what values the variables take. Identities describe patterns that hold universally, making them powerful tools for factorisation.

Difference of Two Squares

The identity $a^{2} - b^{2} = (a + b) (a - b)$ . Any expression in this form — a perfect square minus another perfect square — can be factorised using this pattern.

Perfect Square Trinomial

A trinomial that results from squaring a binomial: $a^{2} + 2 ab + b^{2} = (a + b)^{2}$ or $a^{2} - 2 ab + b^{2} = (a - b)^{2}$ .

Note

You already use the idea of HCF with numbers — for example, the HCF of 12 and 18 is 6. In algebra, we extend this to include variable parts too. The identity-based methods come later in this topic and build on the same foundation.

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