Factorise the following polynomial using synthetic division:
$x^{3} - 10 x^{2} - x + 10$

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Solution

$\begin{matrix} p (x) = x^{3} - 10 x^{2} - x + 10 = 1 - 10 - 1 + 10 = 11 - 11 = 0 \end{matrix}$
$∴ x - 1$ is a factor

$\begin{matrix} x^{2} - 9 x - 10 = x^{2} - 10 x + x - 10 - 10 + 1 = x (x - 10) + 1 (x - 10) = (x - 10) (x + 1) \end{matrix}$
This $(x - 1) (x + 1) (x - 10)$ are the factors.
$∴ x^{3} - 10 x^{2} - x + 10 = (x - 1) (x - 10) (x + 1)$

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