# Complex numbers

Here's the question:

z is a complex, and if $z^5 + z^4 + z^3 + z^2 + z + 1 = 0$ then $z^6=1$.

use this fact to calculate how many answers is there for:

$$z^5 + z^4 + z^3 + z^2 + z + 1 = 0$$

Thanks.

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$z^6 = 1$ if and only if $z^6 - 1 = (z-1)(z^5+z^4+z^3+z^2+z+1) = 0$. So the roots of $z^5+z^4+z^3+z^2+z+1$ consist of the roots of $z^6 = 1$ excluding the root $z = 1$, which leaves $5$ roots.