Complex roots of a 7th degree polynomial
Algebra
Level
4
Consider the equation
\[x^7+x^6+x^5+x^4+x^3+x^2+x+1=0.\]
Let \(x_k\) be the \(k^\text{th}\) distinct root of the equation, and let \(\Re(x_k)\) be the real part of that root. Then
\[\sum_{k=1}^7{\big[\Re(x_k)\big]^2}=\, ?\]
by
Andy Hayes