Factorize : $6 x^{3} - 5 x^{2} - 13 x + 12$

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Solution

Consider the given that $6 x^{3} - 5 x^{2} - 13 x + 12$

Put $x = 1$ and we get,

$6 {(1)}^{3} - 5 {(1)}^{2} - 13 (1) + 12$

$= 6 - 5 - 13 + 12 = 0$

Then, $x = 1$ and $x - 1 = 0$ is a factor.

Now figure above

$6 x^{3} - 5 x^{2} - 13 x + 12 = (x - 1) (6 x^{2} + x - 12) = 0$

$\Rightarrow (x - 1) (6 x^{2} + x - 12) = 0$

$\Rightarrow (x - 1) (6 x^{2} + (9 - 8) x - 12) = 0$

$\Rightarrow (x - 1) (6 x^{2} + 9 x - 8 x - 12) = 0$

$\Rightarrow (x - 1) [3 x (2 x + 3) - 4 (2 x + 3)] = 0$

$\Rightarrow (x - 1) (3 x - 4) (2 x + 3) = 0$

If $x - 1 = 0$ then $x = 1$

If $3 x - 4 = 0$ then $x = \frac{4}{3}$

If $2 x + 3 = 0$ then $x = \frac{- 3}{2}$
Hence, the factor is $1, - \frac{3}{2}$ and $\frac{4}{3}$ .

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