Divide $x^{4} + x^{2} + 1$ by $x^{2} - x + 1$ .

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Solution

Consider $x^{4} + x^{2} + 1 = x^{4} + 2 x^{2} + 1 - x^{2}$
$= (x^{2} + 1)^{2} - x^{2}$
$= (x^{2} + x + 1) (x^{2} - x + 1)$
Dividing $x^{4} + x^{2} + 1$ by $x^{2} - x + 1$ we get
$\frac{x^{4} + x^{2} + 1}{x^{2} - x + 1} = x^{2} + x + 1$

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