Show that (x - 2) is a factor of ${3 x}^{2} - x - 10.$ Hence, factories ${3 x}^{2} - x - 10.$

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Solution

Let
$p (x) = 3 x^{2} - x - 10$ and $g (x) = x - 2$
To prove :
$g (x)$ is a factor of $p (x)$

Proof :
Consider $g (x) = 0 \Rightarrow x - 2 = 0 \Rightarrow x = 2$

Now putting $x = 2$ in $p (x)$ , We get
$p (2) = 3 (2)^{2} - (2) - 10 = 12 - 2 - 10 = 0$
$∴$ By factor theorem, $(x - 2)$ is a factor of $p (x)$
Hence proved.

Now, $3 x^{2} - x - 10$
$= 3 x^{2} - 6 x + 5 x - 10 = 3 x (x - 2) + 5 (x - 2) = (x - 2) (3 x + 5)$

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