Question

What is the formula of ${\left(A+B\right)}^3$?

Answer 1

Find the formula of ${\left(A+B\right)}^3$ :

Distribution property: $a·\left(b+c\right)=a·b+a·c$

${\left(A+B\right)}^3={\left(A+B\right)}^2·\left(A+B\right)$

$\Rightarrow$ ${\left(A+B\right)}^3=\left(A^2+B^2+2AB\right)\left(A+B\right)$ $\left[\because {\left(a+b\right)}^2=a^2+b^2+2ab\right]$

$\Rightarrow$${\left(A+B\right)}^3=\left(A^2+B^2+2AB\right)·A+\left(A^2+B^2+2AB\right)·B$

$\Rightarrow$ ${\left(A+B\right)}^3=A^3+B^{2}A+2A^{2}B+A^{2}B+B^3+2A{B}^2$ $\left[\mathrm{Apply}\mathrm{distribution}\mathrm{property}\right]$

$\Rightarrow$ ${\left(A+B\right)}^3=A^3+B^3+3A^{2}B+3A{B}^2$

$\Rightarrow$ ${\left(A+B\right)}^3=A^3+B^3+3AB\left(A+B\right)$

Hence, the formula of ${\left(A+B\right)}^3$ is $A^3+B^3+3AB\left(A+B\right)$.