Question:
Factorize each of the following expression:
x8 − 1
Solution:
$x^{8}-1$
$=\left(x^{4}\right)^{2}-1^{2}$
$=\left(x^{4}-1\right)\left(x^{4}+1\right)$
$=\left[\left(x^{2}\right)^{2}-1^{2}\right]\left(x^{4}+1\right)$
$=\left(x^{2}-1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$
$=\left(x^{2}-1^{2}\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$
$=(x-1)(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)$