Find the LCM of the following:
$x^{2} - 3 x - 4, x^{2} + 2 x - 24$

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Solution

We know that $L C M$ is the least common multiple.

Factorise $x^{2} - 3 x - 4$ as follows:

$x^{2} - 3 x - 4 = x^{2} - 4 x + x - 4 = x (x - 4) + 1 (x - 4) = (x + 1) (x - 4)$

Now, factorise $x^{2} + 2 x - 24$ as follows:

$x^{2} + 2 x - 24 = x^{2} + 6 x - 4 x - 24 = x (x + 6) - 4 (x + 6) = (x - 4) (x + 6)$

Therefore, the least common multiple between the polynomials $x^{2} - 3 x - 4$ and $x^{2} + 2 x - 24$ is:

$L C M = (x - 4) (x + 1) (x + 6)$

Hence, the $L C M$ is $(x - 4) (x + 1) (x + 6)$ .

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