Question

What is the ${(a-b)}^3$ formula?

Answer 1

Expansion of ${\mathbf{(}\mathbf{a}\mathbf{-}\mathbf{b}\mathbf{)}}^{\mathbf{3}}$:

To derive the formula of ${(a-b)}^3$, multiply $(a-b)$ three times.

$$\begin{gathered} \therefore {(a-b)}^3=(a-b)\times (a-b)\times (a-b) \\ =(a^2-ab-ab+b^2)\times (a-b) \\ =(a^2-2ab+b^2)\times (a-b) \\ =a^3-2a^{2}b+a{b}^2-a^{2}b+2a{b}^2-b^3 \\ =a^3-b^3-3a^{2}b+3a{b}^2 \\ =a^3-b^3-3ab(a-b) \end{gathered}$$

Hence, ${\mathbf{(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}}^{\mathbf{3}}\mathbf{=}{\mathit{a}}^3\mathbf{-}{\mathit{b}}^{\mathbf{3}}\mathbf{-}\mathbf{3}\mathit{a}\mathit{b}\mathbf{(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}$