Find the roots of the summed polynomial 
Find the roots of: $$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$

I got that:
$$\frac{1 - x^8}{1-x} - x^6 - x = 0$$
But that doesnt make it any easier.

$$x^7 + x^5 + x^4 + x^3 + x^2 + 1 = 0$$
$$(x^7 + x^4) + (x^5 + x^3) + x^2 + 1 = 0$$
$$x^4(x^3 + 1) + x^3(x^2 + 1) + x^2 + 1 = 0$$
$$x^4(x^3 + 1) + (x^2 + 1)(x^3 + 1) = 0$$
$$(x^3 + 1)(x^4+x^2 + 1) = 0$$
from there we can continue with factoring
$$x^3+1=x^3+1^3=(x+1)(x^2-x+1)$$
and
$$x^4+x^2 + 1 =(x^2)^2+2x^2+1-x^2=(x^2+1)^2-x^2=$$
$$=(x^2+1-x)(x^2+1+x)$$
