Question

How do you factor $a^3-b^3$?

Answer 1

Factorizing the given expression:

Given, $a^3-b^3$

We know that

${\left(a-b\right)}^3=a^3-3a^{2}b+3a{b}^2-b^3$

$\Rightarrow a^3-b^3={\left(a-b\right)}^3+3a^{2}b-3a{b}^2$ [Rearranging the terms in the equation]

$$\begin{gathered} ={\left(a-b\right)}^3+3ab\left(a-b\right) \\ =\left(a-b\right)\left({\left(a-b\right)}^2+3ab\right) \\ =\left(a-b\right)\left(a^2-2ab+b^2+3ab\right)\left[\because {\left(a-b\right)}^2=a^2-2ab+b^2\right] \\ =\left(a-b\right)\left(a^2+ab+b^2\right) \end{gathered}$$

Hence, $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$.