# How do you factor #x^2 + x + 1 ?

May 4, 2015

To factor ${x}^{2} + x + 1$
use the quadratic formula for roots
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

In this case:
$x = \frac{- 1 \pm \sqrt{{1}^{2} - 4 \left(1\right) \left(1\right)}}{2 \left(1\right)}$

$x = - \frac{1}{2} \pm \frac{\sqrt{- 3}}{2}$

With no Real roots
and thus no factors if we are restricted to Real numbers.

However, if we are allowed to venture into Complex numbers
$x = - \frac{1}{2} \pm \frac{\sqrt{3}}{2} i$

and the factors are
$\left(x + \frac{1}{2} + \frac{\sqrt{3}}{2} i\right) \left(x + \frac{1}{2} - \frac{\sqrt{3}}{2}\right)$