Calculate :
are the roots of the equation . But . Therefore are the roots of the equation . Writing y = 1/x, you see that are the roots of the equation .
If we call the roots of that equation then we know that , and . Now successively compute that
,
,
,
,
.
That looks like a large number, but if you compute the 8th powers of , and , you'll find that the sum comes out right.