How to factorize equations in terms of $w$?

Question

Like we can factorize $x^2 + x + 1$ as (x+$w$)(x+$w^2$).

Where $w$ is cube root of unity

Answer 1

The equation $x^2 + x + 1=0$ has the solutions $w_{1/2}=- \frac{1}{2} \pm \frac{\sqrt{3}}{2}.$

Put $w=w_1$ and show that $w^2=w_2.$

Answer 2

Well, we have $x^3-1 = (x-1)(x^2+x+1)$. So the roots of $x^2+x+1$ are 3rd roots of unity. As exalted Lord Shark said, the factorization is $(x-w)(x-w^2)$, where $w$ is a primitive 3rd root of unity.