Consider the equation: $(x-2)^6 + (x-4)^6 = 64$ This equation has two real roots $a, b$ and two pairs of complex conjugate roots $(p, q)$ and $(r, s)$. Is $p + q = r + s$ or $pq=rs$?
The trivial real roots of the equation are $(2,4)$. My question is, when there are two complex conjugate pairs of a polynomial, do the magnitude of the roots have any sort of relation? Is trivially trying to find them by substituting $x=a+ib$ (with $a=1$ in this case so that there remain no imaginary part on the LHS) the only method? This approach leads to an equation of the sixth degree in $b$.