Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Factorisation — taking expressions back into bracketed form — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise Factorisation in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Factorisation revision-guide angle, while Tutopiya’s Factorisation subtopic page owns the learning resource and the free Factorisation quiz owns the practice.

Factorisation is the reverse of expanding brackets — and one of the highest-value Algebra skills in Cambridge IGCSE Mathematics (0580/0607). You need it to solve quadratic equations, simplify algebraic fractions and prove identities. Examiners test common factors, difference of two squares and quadratic trinomials. This guide explains each type, how examiners phrase the questions, and where to practise.

Key takeaways

• Always check for a common factor first before using other methods.
• Difference of two squares: a² − b² = (a + b)(a − b).
• Quadratic trinomials x² + bx + c: find two numbers that multiply to c and add to b.
• Factorisation unlocks Linear Equations and Inequalities and quadratic solving later in Algebra.

What is Factorisation in Cambridge IGCSE Maths?

Factorisation means writing an expression as a product of factors — usually brackets multiplied together. The expression 6x + 9 factorises to 3(2x + 3). In Cambridge IGCSE Mathematics, you factorise by taking out common factors, recognising the difference of two squares, and factorising quadratics of the form x² + bx + c or ax² + bx + c. Fully factorised means no further common factors remain.

Read the full explanation on Tutopiya’s Factorisation subtopic page before you attempt questions.

The three main factorisation types

Type	When to use	Pattern
Common factor	Every term shares a factor	ax + ay = a(x + y)
Difference of squares	Two terms, both perfect squares, minus sign	4x² − 9 = (2x + 3)(2x − 3)
Quadratic trinomial	Three terms, highest power x²	x² + 5x + 6 = (x + 2)(x + 3)

How to factorise — step by step

1. Check for a common factor in all terms — take it out first.
2. Count the terms — two terms with minus? Try difference of squares. Three terms? Try quadratic.
3. For x² + bx + c, find two numbers p and q with pq = c and p + q = b → (x + p)(x + q).
4. For ax² + bx + c, look for common factor first; otherwise use grouping or trial factors.
5. Expand mentally to verify — multiply your brackets back out.

Test yourself with the free Factorisation quiz.

Factorisation vs expanding: the reverse check

Expanded form	Factorised form	Method
3x² + 6x	3x(x + 2)	Common factor 3x
x² − 16	(x + 4)(x − 4)	Difference of squares
x² − x − 6	(x + 2)(x − 3)	Numbers +2 and −3: product −6, sum −1

Factorisation in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
Factorise fully	Complete factorisation, all common factors removed	”Factorise fully 12x² − 18x.”
Factorise	Same as above unless “fully” omitted on simple cases	”Factorise x² + 7x + 12.”
Write … as a product of factors	Bracketed form	”Write 2x² − 8 as a product of factors.”
Show that	Factorise or expand to prove	”Show that x² − 9 = (x + 3)(x − 3).”
Hence solve	Factorise then use zero product	”Factorise x² − 5x + 6. Hence solve x² − 5x + 6 = 0.”

Worked exam-style stems (how to answer the wording)

1. “Factorise fully 15x²y + 25xy².” Common factor 5xy: 5xy(3x + 5y). Reward: highest common factor of coefficients and variables.
2. “Factorise x² − 49.” Difference of squares: (x + 7)(x − 7). Reward: recognising 49 = 7².
3. “Factorise x² + 2x − 15.” Need product −15, sum 2: 5 and −3. (x + 5)(x − 3). Reward: correct number pair; check by expanding.

Work similar stems on the Algebra topical past paper questions and the Factorisation quiz.

How Factorisation connects to the rest of Algebra

Factorising is the first step in solving many equations — see Linear Equations and Inequalities and later quadratics. It reverses Simplifying Algebraic Expressions. Use the Cambridge IGCSE Maths resource hub for the full Algebra unit.

Common mistakes students make

• Forgetting the common factor and trying to factorise a quadratic that still has a shared factor.
• Wrong signs in the number pair for trinomials — especially when c is negative.
• Stopping at (x + 3)(x + 3) without writing (x + 3)² when asked to factorise fully.
• Confusing difference of squares with trinomials — x² − 9 has two terms only.

When you need more support

If quadratic trinomials keep failing, work through the Simplifying Algebraic Expressions quiz (expanding reinforces factorising) and the Algebra topical past paper questions, then get help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

What does factorise fully mean? Take out all common factors and factorise each part completely — no further factorisation is possible.

How do I factorise x² + bx + c when c is positive and b is positive? Both numbers in the brackets are positive: (x + p)(x + q) where pq = c and p + q = b.

Is factorisation on Paper 1? Yes — many factorisation questions are non-calculator friendly, especially common factor and simple quadratics.

How do I revise Factorisation effectively? Do five of each type (common factor, difference of squares, trinomial), then take the Factorisation quiz.

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