Using factor theorem, factorize each of the following: $x^{3} - 6 x^{2} + 3 x + 10$

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Solution

Let $x = - 1$

So, $f (- 1) = (- 1)^{3} - 6 (- 1)^{2} + 3 (- 1) + 10$
$= - 1 - 6 - 3 + 10$
$= 0$

$∴ (x + 1)$ is a factor
Given $x^{3} - 6 x^{2} + 3 x + 10$
$= x^{3} + x^{2} - 7 x^{2} - 7 x + 10 x + 10$
$= x^{2} (x + 1) - 7 x (x + 1) + 10 (x + 1)$
$= (x + 1) (x^{2} - 7 x + 10)$
$= (x + 1) (x^{2} - 5 x - 2 x + 10)$
$= (x + 1) {x (x - 5) - 2 (x - 4)}$
$= (x + 1) (x - 2) (x - 5)$ .

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