Factorise
$2 x^{3} - 3 x^{2} - 17 x + 30$

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Solution

Let $p (x) = 2 x^{3} - 3 x^{2} - 17 x + 30$
Constant term of p(x) = 30
$∴$ Factors of 30 are $\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30$
By trial, we find that p(2) = 0, so (x – 2) is a factor of p(x).
$[∵ 2 (2)^{3} - 3 (2)^{2} - 17 (2) + 30 = 16 - 12 - 34 + 30 = 0]$
Now, we see that $2 x^{3} - 3 x^{2} - 17 x + 30$

$= (x - 2) (2 x^{2} + x - 15)$
$2 x^{2} + x - 15 = 2 x (x + 3) - 5 (x + 3)$ [By splitting the middle term]
$= (x + 3) (2 x - 5)$
$∴ 2 x^{3} - 3 x^{2} - 17 x + 30 = (x - 2) (x + 3) (2 x - 5)$

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