Factorise the following: $x^{4} + x^{2} + 1$

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Solution

Given: $x^{4} + x^{2} + 1$
$= x^{4} + x^{2} + x^{2} - x^{2} + 1$ , (add and subtract $x^{2}$ )
$= [(x^{2})^{2} + 2 x^{2} + 1] - x^{2}$ , (group the terms)
$= (x^{2} + 1)^{2} - x^{2}$ , (since $a^{2} + 2 a b + b^{2} = (a + b)^{2}$ )
$= [(x^{2} + 1) + x] [(x^{2} + 1) - x]$ , (since $a^{2} - b^{2} = (a + b) (a - b)$ )
$= (x^{2} + x + 1) (x^{2} - x + 1)$ ,

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