Last Updated Sep 27, 2025

Sum / Difference of Two Cubes Simplified Revision Notes for Leaving Cert Mathematics

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Sum / Difference of Two Cubes

The difference of two cubes is an algebraic identity that expresses the difference between two cubed terms.

$$a^3-b^3=(a-b)(a^2+ab+b^2)$$

infoNote

This identity is not in the log tables and must be learnt off.

A cube term can be expressed in the form $a^3$ :

• $27$ can be expressed as $3^3$
• $8y^3$ can be expressed as $(2y)^3$

Example

infoNote

Factorise the following expression $64a^3-125b^3$

Identify cube term and rewrite in the form $a^3-b^3$ :

$$\left(4a \right)^3-\left( 5b\right)^3$$

Apply the identity :

$$\begin{align*} &(4a-5b)((4a)^2+(4a)(5b)+(5b)^2) =\\\\ &(4a-5b)(16a^2+20ab+25b^2) \end{align*}$$

The sum of two cubes can also be factorised using the following identity

$$a^3+b^3=(a+b)(a^2-ab+b^2)$$

infoNote

This identity is not in the log tables and must be learnt off.

Example

infoNote

Factorise the following expression $27p^9+q^3$

Identify cube term and rewrite in the form $a^3+b^3$ :

$$\left(3p^3 \right)^3+\left( q\right)^3$$

Apply the identity :

$$\begin{align*} &(3p^3 +q)\left(\left(3p^3 \right)^2-(3p^3 )(q)+(q)^2\right) =\\\\ &(3p^3 +q)\left(9p^6-3p^3q+q^2\right) \end{align*}$$

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