Factorize: $1 + x^{4} + x^{8}$

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Solution

$1 + x^{4} + x^{8}$
$= 1 + x^{4} + x^{8} + x^{4} - x^{4}$
$= 1 + 2 x^{4} + x^{8} - x^{4}$
$= (1 + x^{4})^{2} - x^{4}$
$= (1 + x^{4})^{2} - (x^{2})^{2}$
$= (1 + x^{4} + x^{2}) (1 + x^{4} - x^{2})$
$= (1 + x^{2} + x^{2} + x^{4} - x^{2}) (1 + x^{4} - x^{2})$
$= (1 + 2 x^{2} + x^{4} - x^{2}) (1 + x^{4} - x^{2})$
$= [(1 + x^{2})^{2} - x^{2}] [1 + x^{4} - x^{2}]$
$= (1 + x^{2} + x) (1 + x^{2} - x) (1 - x^{2} + x^{4})$
$= (1 + x + x^{2}) (1 - x + x^{2}) (1 - x^{2} + x^{4})$

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