Question

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

Answer 1

Determine the formula$a^3-b^3$.

We know that, for any real number$a\mathrm{and}b$,

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

Rewrite the formula:

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

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

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

Therefore, the formula is $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$.