# How do you factor x^15 - 1/64?

Apr 28, 2016

${x}^{15} - \frac{1}{64} = \left({x}^{5} - \frac{1}{4}\right) \left({x}^{10} + \frac{1}{4} {x}^{5} + \frac{1}{16}\right)$

#### Explanation:

${x}^{15} - \frac{1}{64} = {\left({x}^{5}\right)}^{3} - {\left(\frac{1}{4}\right)}^{3}$ and hence using identity

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

${\left({x}^{5}\right)}^{3} - {\left(\frac{1}{4}\right)}^{3} = \left({x}^{5} - \frac{1}{4}\right) \left({\left({x}^{5}\right)}^{2} + \frac{1}{4} {x}^{5} + {\left(\frac{1}{4}\right)}^{2}\right)$

= $\left({x}^{5} - \frac{1}{4}\right) \left({x}^{10} + \frac{1}{4} {x}^{5} + \frac{1}{16}\right)$