Factorise :
(i) $x^{3} - 2 x^{2} - x + 2$
(ii) $x^{3} + 13 x^{2} + 32 x + 20$

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Solution

(i) $x^{3} - 2 x^{2} - x + 2$
$= x^{2} (x - 2) - 1 (x - 2)$
$= (x^{2} - 1) (x - 2)$
$= (x - 1) (x + 1) (x - 2)$
(ii) $x^{3} + 13 x^{2} + 32 x + 20$
$= x^{3} + 12 x^{2} + x^{2} + 12 x + 20 x + 20$
$= x^{3} + x^{2} + 12 x^{2} + 12 x + 20 x + 20$
$= x^{2} (x + 1) + 12 x (x + 1) + 20 (x + 1)$
$= (x^{2} + 12 x + 20) (x + 1)$
$= (x^{2} + 10 x + 2 x + 20) (x + 1)$
$= {x (x + 10) + 2 (x + 10)} (x + 1)$
$= (x + 10) (x + 2) (x + 1)$ .

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