Question

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

Answer 1

Find the formula for${\left(a-b\right)}^3$.

The expression ${\left(a-b\right)}^3$ can be written as follows:

${\left(a-b\right)}^3=\left(a-b\right){\left(a-b\right)}^2\dots \left(1\right)$

Substitute the value ${\left(a-b\right)}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{a}\mathit{b}$into $\left(1\right)$:

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

Use the distributive property: $a\left(b+c\right)=ab+ac$:

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

Hence, the formula is $\begin{array}{rcl}{\left(a-b\right)}^3& =& a^3-b^3-3ab(a-b)\end{array}$.