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

1 Answer

May 4, 2015

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

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

$x = - \frac{1}{2} \pm \frac{\sqrt{- 3}}{2}$ $x = -1/2 +-sqrt(-3)/2$

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
$x = - \frac{1}{2} \pm \frac{\sqrt{3}}{2} i$ $x= -1/2 +-sqrt(3)/2i$

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

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