If a complex number \(\alpha\) is a common root of
\[
x^{2026}+x^{1964}+1=0
\]
and
\[
x^3+2x^2+2x+1=0,
\]
then the sum of the complex roots of the equation
\[
z^3=\alpha^3
\]
is
Show Hint
The cube roots of unity are
\[
1,\quad
\omega,\quad
\omega^2,
\]
where
\[
\boxed{1+\omega+\omega^2=0.}
\]
Hence, the sum of the two non-real cube roots is
\[
\boxed{\omega+\omega^2=-1.}
\]