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

= (x^{2} + 1 + x) (x^{2} + 1 - x) (x^{2} - x^{2} + 7)

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= (x^{3} + 1 + x) (x^{2} + 1 - x) (x^{3} - x^{2} + 4)

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= (x^{3} + 1 + x) (x^{2} + 1 - x) (x^{4} - x^{2} + 2)

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

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Solution

The correct option is D $= (x^{2} + 1 + x) (x^{2} + 1 - x) (x^{4} - x^{2} + 1)$
$x^{8} + x^{4} + 1 = x^{8} + {2 x}^{4} + 1 - x^{4}$
$= {(x^{4} + 1)}^{2} - {(x^{2})}^{2} = (x^{4} + 1 + x^{2}) (x^{4} + 1 - x^{2})$
$= [{{(x}^{2} + 1)}^{2} - x^{2}] (x^{4} - x^{2} + 1)$
$= (x^{2} + 1 + x) (x^{2} + 1 - x) (x^{4} - x^{2} + 1)$

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