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

May 16, 2018

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

#### Explanation:

${a}^{3} {b}^{6} - {b}^{3}$

${b}^{6} = {b}^{3} \cdot {b}^{3}$

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

factorizing;

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

May 16, 2018

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

#### Explanation:

We want to simplify ${a}^{3} {b}^{6} - {b}^{3}$. The first thing to do is to factor out ${b}^{3}$ to give ${b}^{3} \left({a}^{3} {b}^{3} - 1\right)$.

If we look carefully, we can see that ${b}^{3} \left({a}^{3} {b}^{3} - 1\right) = {b}^{3} \left({\left(a b\right)}^{3} - {1}^{3}\right)$. So now we can use a formula to simplify even more:

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

So

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