$\begin{array}{l}\phantom{\phantom{;}z^{2}+1;}{\phantom{;}z^{3}\phantom{-;x^n}-z\phantom{;}\phantom{-;x^n}}\\\phantom{;}z^{2}+1\overline{\smash{)}\phantom{;}z^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{\phantom{;}z^{2}+1;}\underline{-z^{5}\phantom{-;x^n}-z^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-z^{5}-z^{3};}-z^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\\\phantom{\phantom{;}z^{2}+1-;x^n;}\underline{\phantom{;}z^{3}\phantom{-;x^n}+z\phantom{;}\phantom{-;x^n}}\\\phantom{;\phantom{;}z^{3}+z\phantom{;}-;x^n;}\phantom{;}z\phantom{;}\phantom{-;x^n}\\\end{array}$

$z^{3}-z+\frac{z}{z^2+1}$