The HCF $h (y) = 2 y (y + 2)$ and the LCM $m (y) = 24 y {(y + 2)}^{2} (y - 2)$ for two polynomials $p (y)$ and $q (y)$ is known. If $p (y) = 8 y^{3} + 32 y^{2} + 32 y$ and if $h (y) m (y) = - p (y) q (y)$ , find $q (y)$ .

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Solution

HCF $h (y) = 2 y (y + 2)$ and LCM $m (y) = 24 y {(y + 2)}^{2} (y - 2)$ for $p (y)$ and $q (y)$
Now, $p (y) = 8 y^{3} + 32 y^{2} + 32 y$
$= 8 y (y^{2} + 4 y + 4)$
$= 8 y (y^{2} + 2 y + 2 y + 4)$
$= 8 y [y (y + 2) + 2 (y + 2)]$
$= 8 y (y + 2)^{2}$
Also, $h (y) m (y) = - p (y) q (y)$
$⟹ 2 y (y + 2) 24 y (y + 2)^{2} (y - 2) = - 8 y (y + 2)^{2} \cdot q (y)$
$⟹ 6 y (y + 2) (y - 2) = - q (y)$
$⟹ q (y) = - 6 y (y^{2} - 2^{2})$

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