Multiply both sides by \(({z}^{2}+1)({z}^{2}+2)\).
\[z({z}^{2}+1)({z}^{2}+2)=1\]
Expand.
\[{z}^{5}+2{z}^{3}+{z}^{3}+2z=1\]
Simplify \({z}^{5}+2{z}^{3}+{z}^{3}+2z\) to \({z}^{5}+3{z}^{3}+2z\).
\[{z}^{5}+3{z}^{3}+2z=1\]
Move all terms to one side.
\[{z}^{5}+3{z}^{3}+2z-1=0\]
No root was found algebraically. However, the following root(s) were found by numerical methods.
\[z=0.399372\]
z=0.39937210083008
