Question

What is the formula for ${\left(a+b\right)}^3$?

Answer 1

Determine the formula of ${\left(a+b\right)}^3$.

Rewrite the expression ${\left(a+b\right)}^3$:

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

Apply the identity ${\left(a+b\right)}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}$ and the distributive property, $a\left(b+c\right)=ab+ac$:

$\begin{array}{rcl}{\left(a+b\right)}^3& =& \left(a+b\right)\left(a^2+b^2+2ab\right)\\ & \Rightarrow & {\left(a+b\right)}^3=a^3+a{b}^2+2a^{2}b+b{a}^2+b^3+2a{b}^2\\ & \Rightarrow & {\left(a+b\right)}^3=a^3+b^3+3a^{2}b+3a{b}^2\end{array}$

Therefore, the formula is $\begin{array}{rcl}{\left(a+b\right)}^3& =& a^3+b^3+3a^{2}b+3a{b}^2\end{array}$.