The L.C.M. and H.C.F. of two polynomials p(x) | vedclass

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(C) We know that for any two polynomials $p(x)$ and $q(x)$:$p(x) 	imes q(x) = L.C.M. 	imes H.C.F.$Given:$p(x) = 4 x^{2}(x^{2}-a^{2})$$L.C.M. = 36 x^

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$9 x^{3}(x^{3}-a^{3})$

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(C) We know that for any two polynomials $p(x)$ and $q(x)$:$p(x) 	imes q(x) = L.C.M. 	imes H.C.F.$Given:$p(x) = 4 x^{2}(x^{2}-a^{2})$$L.C.M. = 36 x^{3}(x+a)(x^{3}-a^{3})$$H.C.F. = x^{2}(x-a)$Substituting these values into the formula:$4 x^{2}(x^{2}-a^{2}) 	imes q(x) = 36 x^{3}(x+a)(x^{3}-a^{3}) 	imes x^{2}(x-a)$Since $(x^{2}-a^{2}) = (x-a)(x+a)$,we can write:$4 x^{2}(x-a)(x+a) 	imes q(x) = 36 x^{5}(x+a)(x-a)(x^{3}-a^{3})$$q(x) = \frac{36 x^{5}(x+a)(x-a)(x^{3}-a^{3})}{4 x^{2}(x+a)(x-a)}$$q(x) = 9 x^{3}(x^{3}-a^{3})$

The $L.C.M.$ and $H.C.F.$ of two polynomials $p(x)$ and $q(x)$ are $36 x^{3}(x+a)(x^{3}-a^{3})$ and $x^{2}(x-a)$ respectively. If $p(x)=4 x^{2}(x^{2}-a^{2})$,find $q(x)$.

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