Question

Find the product: $\left(p^2-q^2\right)\left(2p+q\right)$

Answer 1

Simplify using the rules of binomial products:

The given expression is $\left(p^2-q^2\right)\left(2p+q\right)$.

It can be simplified as follows,

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

$\Rightarrow \left(p^2-q^2\right)\left(2p+q\right)=p^2\times 2p+p^2\times q-q^2\times 2p-q^2\times q$

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

$\Rightarrow \left(p^2-q^2\right)\left(2p+q\right)=2p^3+p^{2}q-2p{q}^2-q^3$

$\Rightarrow \left(p^2-q^2\right)\left(2p+q\right)=2p^3-q^3+p^{2}q-2p{q}^2$

Hence, the product of $\left(p^2-q^2\right)\left(2p+q\right)$ is $2p^3-q^3+p^{2}q-2p{q}^2$.