Find the LCM of $3 x^{2} - 75, 2 x^{3} + 250$

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Solution

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

Factorise $3 x^{2} - 75$ as follows:

$3 x^{2} - 75 = 3 (x^{2} - 25) = 3 (x^{2} - 5^{2}) = 3 (x + 5) (x - 5)$ (using identity $a^{2} - b^{2} = (a + b) (a - b)$ )

Now, factorise $2 x^{3} + 250$ as follows:

$2 x^{3} + 250 = 2 (x^{3} + 125)$

Therefore, the least common multiple between the polynomials $3 x^{2} - 75$ and $2 x^{3} + 250$ is:

$L C M = (2 \times 3) (x + 5) (x^{3} + 125) = 6 (x + 5) (x^{3} + 125)$ .

Hence, the $L C M$ is $6 (x + 5) (x^{3} + 125)$ .

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