# How do you factor (r^3 - 1)?

Jul 31, 2016

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

#### Explanation:

One can either use the identity $\left({a}^{3} - {b}^{3}\right) = \left(a - b\right) \left({a}^{2} + a b + {b}^{2}\right)$, so that

${r}^{3} - 1 = {r}^{3} - {1}^{3} = \left(r - 1\right) \left({r}^{2} + r \times 1 + {1}^{2}\right) = \left(r - 1\right) \left({r}^{2} + r + 1\right)$

We can also factorize following way

${r}^{3} - 1$

= ${r}^{3} - {r}^{2} + {r}^{2} - r + r - 1$

= ${r}^{2} \left(r - 1\right) + r \left(r - 1\right) + 1 \left(r - 1\right)$

= $\left(r - 1\right) \left({r}^{2} + r + 1\right)$