Factorise the following polynomials
$x^{3} - 5 x + 4$

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Solution

$x^{3} - 5 x + 4 = x^{3} - x - 4 x + 4 = x (x^{2} - 1) - 4 (x - 1) = x (x + 1) (x - 1) - 4 (x - 1) = (x - 1) (x^{2} + x - 4)$
The zeroes of $x^{2} + x - 4$ are $\frac{- 1 \pm \sqrt{1 + 16}}{2} i . e \frac{- 1 \pm \sqrt{17}}{2}$
So $x^{3} - 5 x + 4 = (x - 1) (x + \frac{1 - \sqrt{17}}{2}) (x + \frac{1 + \sqrt{17}}{2})$

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