$\begin{array}{l}\phantom{\phantom{;}y\phantom{;}+1;}{\phantom{;}y^{7}-y^{6}+y^{5}-y^{4}+y^{3}-y^{2}+y\phantom{;}-1\phantom{;}\phantom{;}}\\\phantom{;}y\phantom{;}+1\overline{\smash{)}\phantom{;}y^{8}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}y\phantom{;}+1;}\underline{-y^{8}-y^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-y^{8}-y^{7};}-y^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n;}\underline{\phantom{;}y^{7}+y^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}y^{7}+y^{6}-;x^n;}\phantom{;}y^{6}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n;}\underline{-y^{6}-y^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-y^{6}-y^{5}-;x^n-;x^n;}-y^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n-;x^n;}\underline{\phantom{;}y^{5}+y^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}y^{5}+y^{4}-;x^n-;x^n-;x^n;}\phantom{;}y^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n-;x^n-;x^n;}\underline{-y^{4}-y^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;-y^{4}-y^{3}-;x^n-;x^n-;x^n-;x^n;}-y^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}y^{3}+y^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;;;\phantom{;}y^{3}+y^{2}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}y^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{-y^{2}-y\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;;;-y^{2}-y\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}-y\phantom{;}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}y\phantom{;}+1-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}y\phantom{;}+1\phantom{;}\phantom{;}}\\\phantom{;;;;;;;\phantom{;}y\phantom{;}+1\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2\phantom{;}\phantom{;}\\\end{array}$

$y^{7}-y^{6}+y^{5}-y^{4}+y^{3}-y^{2}+y-1+\frac{2}{y+1}$