IGCSE · Practice questions · Factorising expressions

Take it out, then look inside.

Six original Cambridge-style questions on factorising fully, from common factors to quadratics, the difference of two squares and grouping.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Common factor first, then a pattern.

HCF: biggest number, and each letter to its lowest power.
\(x^{2} + bx + c\): two numbers that multiply to \(c\) and add to \(b\).
\(ax^{2} + bx + c\): numbers that multiply to \(ac\) and add to \(b\); split and group.
Difference of two squares: \(p^{2} - q^{2} = (p - q)(p + q)\).
Check: expand your answer and look for anything still common.

Recall

[2 marks]

Factorise fully (a) \(8y - 20\) (b) \(5p^{2} + 15pq\).

(a) \(4(2y - 5)\) ✓
(b) \(5p(p + 3q)\) ✓

Calculation

[2 marks]

Factorise fully \(14a^{2}b^{3} + 21ab^{2}\).

HCF \(= 7ab^{2}\) ✓
\(7ab^{2}(2ab + 3)\) ✓

Calculation

[4 marks]

(Extended) Factorise (a) \(x^{2} - 3x - 28\) (b) \(3x^{2} - 10x + 8\).

(a) Product \(-28\), sum \(-3\): \(-7\) and \(4\). \((x - 7)(x + 4)\) ✓
(b) \(ac = 24\), numbers \(-6\) and \(-4\): \(3x^{2} - 6x - 4x + 8\) ✓
\(= 3x(x - 2) - 4(x - 2) = (3x - 4)(x - 2)\) ✓

Calculation

[4 marks]

(Extended) Factorise completely (a) \(25x^{2} - 4y^{2}\) (b) \(3x^{3} - 12x\).

(a) \((5x)^{2} - (2y)^{2} = (5x - 2y)(5x + 2y)\) ✓
(b) \(3x(x^{2} - 4) = 3x(x - 2)(x + 2)\) ✓

Calculation

[2 marks]

(Extended) Factorise \(xy + 3x - 2y - 6\).

\(x(y + 3) - 2(y + 3)\) ✓
\(= (y + 3)(x - 2)\) ✓

Proof

[4 marks]

(Extended)

(a) Without a calculator, use factorising to work out \(87^{2} - 13^{2}\).

(b) \(n\) is a whole number. Show that \((2n + 1)^{2} - (2n - 1)^{2}\) is always a multiple of 8.

(a) \(87^{2} - 13^{2} = (87 - 13)(87 + 13) = 74 \times 100 = 7400\) ✓
(b) Difference of two squares: \([(2n + 1) - (2n - 1)][(2n + 1) + (2n - 1)]\) ✓
\(= 2 \times 4n = 8n\) ✓
\(8n\) is 8 times a whole number, so it is a multiple of 8. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.