# How do you factor a^3*b^6 - b^3?

May 20, 2016

${a}^{3} \cdot {b}^{6} - {b}^{3} = {b}^{3} \cdot \left(a \cdot b - 1\right) \left(1 + a \cdot b + {a}^{2} \cdot {b}^{2}\right)$

#### Explanation:

First, ${a}^{3} \cdot {b}^{6} - {b}^{3} = \left({a}^{3} \cdot {b}^{3} - 1\right) \cdot {b}^{3}$.
Now there is a polynomial identity that can help us and which is
$\frac{{x}^{n + 1} - 1}{x - 1} = 1 + x + {x}^{2} + {x}^{3} + \ldots + {x}^{n}$
then
${a}^{3} \cdot {b}^{3} - 1 = \left(a \cdot b - 1\right) \left(1 + a \cdot b + {a}^{2} \cdot {b}^{2}\right)$. Finally
${a}^{3} \cdot {b}^{6} - {b}^{3} = {b}^{3} \cdot \left(a \cdot b - 1\right) \left(1 + a \cdot b + {a}^{2} \cdot {b}^{2}\right)$