Factorize $x^{5} - x$

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Solution

$x^{5} - x = x (x^{4} - 1) = x [(x^{2})^{2} - (1)^{2}]$
$= x (x^{2} + 1) (x^{2} - 1) = x (x^{2} + 1) [(x)^{2} - (1)^{2}]$
$= x (x^{2} + 1) (x + 1) (x - 1)$

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