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LCM of Polynomials

The HCF of tw...

Question

The HCF of two polynomials $p (x)$ and $q (x)$ is $2 x (x + 2)$ and LCM is $24 x (x + 2)^{2} (x - 2)$ . If $p (x) = 8 x^{3} + 32 x^{2} + 32$ , then what is $q (x)$ equal to?

4 x^{3} - 16 x

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6 x^{3} - 24 x

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12 x^{3} + 24 x

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12 x^{3} - 24 x

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Solution

The correct option is B $6 x^{3} - 24 x$
Given, HCF of $p (x)$ and $q (x)$ is $2 x (x + 2)$ and LCM $= 24 x (x + 2)^{2} (x - 2)$ , $p (x) = 8 x^{3} + 32 x^{2} + 32$
$q (x) = \frac{HCF \times LCM}{p (x)}; p (x) = 8 x (x + 2)^{2}$
$= \frac{2 x (x + 2) \times 24 x (x + 2)^{2} (x - 2)}{8 x (x + 2)^{2}}$
$= 6 x (x + 2) (x - 2)$
$= 6 x (x^{2} - 4)$
$= 6 x^{3} - 24 x$

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