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.

$\dfrac{x^7 + x^5 + x^4 + x^3 + x^2 + 1}{x+1}=x^6-x^5+2x^4-x^3+2x^2-x+1$
$=x^6-x^5+x^4+(x^4-x^3+x^2)+(x^2-x+1)$
$=x^4(x^2-x+1)+x^2(x^2-x+1)+(x^2-x+1)$
$=(x^2-x+1)[x^4+x^2+1]$
Now $x^4+x^2+1=(x^2+1)^2-(x)^2=\cdots$
