Factorise $x^{3} - x^{2} + a x - a + x - 1$ .

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Solution

We group the expression as
$x^{3} - x^{2} + a x - a + x - 1 = (x^{3} - x^{2}) + (a x - a) + (x - 1)$
$= x^{2} (x - 1) + a (x - 1) + 1 (x - 1) = (x - 1) (x^{2} + a + 1)$
Thus $x - 1$ and $x^{2} + a + 1$ are the factors

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