The G.C.D. and L.C.M. of two polynomials p(x) | vedclass

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Question

(A) We know that for any two polynomials $p(x)$ and $q(x)$,the product of the polynomials is equal to the product of their $H.C.F.$ and $L.C.M.$$p(x)

Vedclass · Accepted Answer

$3x^2(x^2-a^2)$

Vedclass · Answer

(A) We know that for any two polynomials $p(x)$ and $q(x)$,the product of the polynomials is equal to the product of their $H.C.F.$ and $L.C.M.$$p(x) 	imes q(x) = H.C.F. 	imes L.C.M.$Given:$H.C.F. = x(x+a)$$L.C.M. = 12x^2(x+a)(x^2-a^2)$$p(x) = 4x(x+a)^2$Substituting the values into the formula:$q(x) = \frac{H.C.F. 	imes L.C.M.}{p(x)}$$q(x) = \frac{x(x+a) 	imes 12x^2(x+a)(x^2-a^2)}{4x(x+a)^2}$$q(x) = \frac{12x^3(x+a)^2(x^2-a^2)}{4x(x+a)^2}$$q(x) = 3x^2(x^2-a^2)$

The $G.C.D.$ and $L.C.M.$ of two polynomials $p(x)$ and $q(x)$ are $x(x+a)$ and $12x^2(x+a)(x^2-a^2)$ respectively. If $p(x) = 4x(x+a)^2$,find $q(x)$.

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