Answer
$(z+1)(z^2-z+1)$
Work Step by Step
Using $a^3+b^3=(a+b)(a^2-ab+b^2)$ or the factoring of 2 cubes, the factored form of the given expression, $
z^3+1
,$ is
\begin{array}{l}
(z+1)[(z)^2-(z)(1)+(1)^2]
\\\\=
(z+1)(z^2-z+1)
.\end{array}