If re^{i theta} is a root of z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1 = 0, where r > 0 and 0 <= theta < 2pi, then find the sum of all possible values of theta.
+288 z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z^1 + 1 =
(z^9-1)/(z-1) = 0.
The roots will all have a magnitutude of 1 on the complex plane so r will always be 1.
The 9th roots of unity(excluding 1) are:
e^{2pi/9i}
e^{4pi/9i}
....
e^{16pi/9i}
Adding up all of the values of theta gives us: 8pi.
