Identify factor of $6 x^{3} + 5 x^{2} - 2 x - 1$

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Solution

Let $P (x) = 6 x^{3} + 5 x^{2} - 2 x - 1$
Now, $P (- 1) = - 6 + 5 + 2 - 1 = 0$
Therefore, at $x = - 1, x + 1 = 0$
Hence, $(x + 1)$ is one of the factor of equation $6 x^{3} + 5 x^{2} - 2 x - 1$
Which can be written as $(x + 1) (6 x^{2} - x - 1)$
$= (x + 1) [6 x^{2} - 3 x + 2 x - 1]$
$= (x + 1) [3 x (2 x - 1) + 1 (2 x - 1)]$
$= (x + 1) (3 x + 1) (2 x - 1)$
Hence the factor of $P (x) = 6 x^{3} + 5 x^{2} - 2 x - 1$ are $(x + 1), (3 x + 1), (2 x - 1)$

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