The HCF of two polynomials 𝑃(𝑥) and 𝑞(𝑥) is 2𝑥(𝑥 + 2) and L.C.M is $24 x (x + 2)^{2} (x - 2)$ . If $P (x) = 8 x^{3} + 32 x^{2} + 32 x$ , then 𝑞(𝑥) is

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$6 x^{3} - 24 x$

$12 x^{3} + 24 x$

$12 x^{3} - 24 x$

$4 x^{3} - 6 x$

answer is B.

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Detailed Solution

$q (x) = \frac{H C F \times L C M}{P (x)}$

$P (x) = 8 x^{3} + 32 x^{2} + 32 x = 8 x (x^{2} + 4 x + 4) = 8 x (x + 2)^{2}$

$∴ q (x) = \frac{2 x (x + 2) \times 24 x (x + 2)^{2} (x - 2)}{8 x (x + 2)^{2}}$

$= 6 x (x + 2) (x - 2)$

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

$= 6 x^{3} - 24 x$

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