Question

How do you factorize $x^4-y^4$?

Answer 1

Factorize using the common factor method

The given expression is $x^4-y^4$.

It can be factorized as follows,

$x^4-y^4=x^{2\times 2}-y^{2\times 2}$

$\Rightarrow x^4-y^4={\left(x^2\right)}^2-{\left(y^2\right)}^2$ $\left[\because a^{m\times n}={\left(a^m\right)}^n\right]$

$\Rightarrow x^4-y^4=\left(x^2-y^2\right)\left(x^2+y^2\right)$ $\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right]$

$\Rightarrow x^4-y^4=\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)$ $\left[\because a^2-b^2=\left(a-b\right)\left(a+b\right)\right]$

Hence, the factors of $x^4-y^4$ obtained by using the common factor method is $\left(x-y\right)\left(x+y\right)\left(x^2+y^2\right)$.