Question

Find the product: $\left(a^2+b\right)\left(a+b^2\right)$

Answer 1

Simplify using the rules of binomial products:

The given expression is $\left(a^2+b\right)\left(a+b^2\right)$.

It can be simplified as follows,

$\left(a^2+b\right)\left(a+b^2\right)=a^2\left(a+b^2\right)+b\left(a+b^2\right)$ $\mathrm{[}\mathrm{\because }\mathrm{}\mathrm{(}\mathrm{a}\mathrm{+}\mathrm{b}\mathrm{)}\mathrm{(}\mathrm{c}\mathrm{+}\mathrm{d}\mathrm{)}\mathrm{=}\mathrm{a}\mathrm{(}\mathrm{c}\mathrm{+}\mathrm{d}\mathrm{)}\mathrm{+}\mathrm{b}\mathrm{(}\mathrm{c}\mathrm{+}\mathrm{d}\mathrm{)}\mathrm{]}$

$\Rightarrow \left(a^2+b\right)\left(a+b^2\right)=a^2\times a+a^2\times b^2+b\times a+b\times b^2$

$\Rightarrow \left(a^2+b\right)\left(a+b^2\right)=a^{2+1}+a^2{b}^2+ab+b^{1+2}$ $\left[\because a^m\times a^n=a^{m+n}\right]$

$\Rightarrow \left(a^2+b\right)\left(a+b^2\right)=a^3+a^2{b}^2+ab+b^3$

$\Rightarrow \left(a^2+b\right)\left(a+b^2\right)=a^3+b^3+a^2{b}^2+ab$

Hence, the product of $\left(a^2+b\right)\left(a+b^2\right)$ is $a^3+b^3+a^2{b}^2+ab$.