Question

How do you factor completely $a^4-b^4$?

Answer 1

Factoring the given expression:

First, use the formula of $a^2-b^2=(a+b)(a-b)$:

$\begin{array}{rcl}{a}^4-b^4& =& {\left(a^2\right)}^2-{\left(b^2\right)}^2\\ & =& {\left(a^2\right)}^2-{\left(b^2\right)}^2\\ & =& \left(a^2+b^2\right)\left(a^2-b^2\right)\end{array}$

Again, use the formula of $a^2-b^2$:

$a^4-b^4=\left(a^2+b^2\right)\left(a+b\right)\left(a-b\right)$

Hence, the factors of $a^4-b^4$ is $\left(a^2+b^2\right)\left(a+b\right)\left(a-b\right)$.