Factorize each of the following expression:

Question:

Factorize each of the following expression:
x8 − 1

Solution:

$x^{8}-1$

$=\left(x^{4}\right)^{2}-1^{2}$

$=\left(x^{4}-1\right)\left(x^{4}+1\right)$

$=\left[\left(x^{2}\right)^{2}-1^{2}\right]\left(x^{4}+1\right)$

$=\left(x^{2}-1\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$

$=\left(x^{2}-1^{2}\right)\left(x^{2}+1\right)\left(x^{4}+1\right)$

$=(x-1)(x+1)\left(x^{2}+1\right)\left(x^{4}+1\right)$

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