Factorize $x^{3} - 3 x^{2} - 10 x + 24$

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Solution

Let $p (x) = x^{3} - 3 x^{2} - 10 x + 24$ .
Since $p (1) \neq 0$ and $p (- 1) \neq 0$ , neither $x + 1$ nor $x + 1$ is a factor of $p (x)$ . Therefore, we have to search for different values of x by trial and error method.
When x=2, p(2)=0. Thus, x-2 is a factor of p(x).
To find the other factors, let us use the synthetic division.
$∴$ The factor is $x^{2} - x - 12$ .
Now, $x^{2} - x - 12 = x^{2} - 4 x + 3 x - 12 = (x - 4) (x + 3)$
Hence, $x^{3} - 3 x^{2} - 10 x + 24 = (x - 2) (x + 3) (x - 4)$

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