Divide the following algebraic identity:$6 a^{2} b^{3}+12 a b^{4}-24 a^{2} b^{5}$ by $6 a^{2} b^{2}$

Given :

The given expressions are $6 a^{2} b^{3}+12 a b^{4}-24 a^{2} b^{5}$ and $6 a^{2} b^{2}$.

To do :

We have to divide $6 a^{2} b^{3}+12 a b^{4}-24 a^{2} b^{5}$ by $6 a^{2} b^{2}$.

Solution :

$6 a^{2} b^{3}+12 a b^{4}-24 a^{2} b^{5}$ by $6 a^{2} b^{2}$

$\Rightarrow \frac{6 a^{2} b^{3}+12 a b^{4}-24 a^{2} b^{5}}{6 a^{2} b^{2}}$

Take $6ab^2$ as common from the numerator,

$\Rightarrow \frac{6ab^2(a+2b-4ab^2)}{6 a^{2} b^{2}} $

$\Rightarrow \frac{b}{a}(a+2b-4ab^2)$               $[\frac{6ab^2}{6 a^{2} b^{2}}=\frac{b}{a}]$

Therefore, the resultant algebraic expression is $\frac{b}{a}(a+2b-4ab^2)$.

$6a^2b^3+12ab^4-24a^2b^5$