The H.C.F. (Highest Common Factor) of two pol | vedclass

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(A) We know that for any two polynomials $p(y)$ and $q(y)$,the relationship is: $p(y) 	imes q(y) = H.C.F. 	imes L.C.M.$Given:$H.C.F. = (y-7)$$L.C.M.

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$y^{2}-12y+35$

Vedclass · Answer

(A) We know that for any two polynomials $p(y)$ and $q(y)$,the relationship is: $p(y) 	imes q(y) = H.C.F. 	imes L.C.M.$Given:$H.C.F. = (y-7)$$L.C.M. = y^{3}-10y^{2}+11y+70$$p(y) = y^{2}-5y-14 = (y-7)(y+2)$Substituting the values into the formula:$(y-7)(y+2) 	imes q(y) = (y-7) 	imes (y^{3}-10y^{2}+11y+70)$$q(y) = \frac{(y-7)(y^{3}-10y^{2}+11y+70)}{(y-7)(y+2)}$$q(y) = \frac{y^{3}-10y^{2}+11y+70}{y+2}$Performing polynomial division of $(y^{3}-10y^{2}+11y+70)$ by $(y+2)$:$y^{3}+2y^{2} - 12y^{2}-24y + 35y+70 = y^{2}(y+2) - 12y(y+2) + 35(y+2)$$= (y^{2}-12y+35)(y+2)$Thus,$q(y) = y^{2}-12y+35$.

The $H.C.F.$ (Highest Common Factor) of two polynomials is $(y-7)$ and their $L.C.M.$ is $y^{3}-10y^{2}+11y+70$. If one of the polynomials is $y^{2}-5y-14$,find the other.

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