# How do you factor a^6 + 1?

Apr 14, 2015

We can modify the expression to use the Sum of Cubes formula to factorise it.

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

The formula says :color(blue)(x^3 + y^3 = (x + y)(x^2-xy+y^2)

Here, $x$ is ${a}^{2}$ and $y$ is $1$

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

color(green)(= (a^2+1)(a^4 - a^2+1)

As none of the factors can be factorised further, this becomes the Factorised form of ${a}^{6} + 1$