Resolve $x^{8} + x^{4} + 1$ in to factors.

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Solution

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

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

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

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

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

$= ({(x^{2})}^{2} + 2 x^{2} + 1 - x^{2}) ({(x^{2})}^{2} - 2 x^{2} + 1 + x^{2})$

$= ({(x^{2} + 1)}^{2} - x^{2}) ({(x^{2} - 1)}^{2} + x^{2})$

$= (x^{2} + x + 1) (x^{2} - x + 1) ({(x^{2} - 1)}^{2} + x^{2})$

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