Complex solutions of $(x+1)^{10}+(x-1)^{10}=0$ Solve $$(x+1)^{10}+(x-1)^{10}=0$$
What method is available to solve this equation?

For $x\in \mathbb{R}$ $(x+1)^{10}+(x-1)^{10}=0$ don't have any solution. This is true by the argument Dr.Sonnhard Graubner.
So let's rewrite question, $$(z+1)^{10}+(z-1)^{10}=0\text{ for }z\in \mathbb{C}.$$
Then suppose if $(z-1)^{10}=0,$then $\implies z=1$ , but $(1+1)^{10}\not=0$ hence $(z-1)^{10}\not=0$.
\begin{gather}
\Rightarrow\frac {(z+1)^{10}}{(z-1)^{10}}+1=0 \\ \frac{z+1}{z-1}=\sqrt[10]{-1}
\\
\end{gather}
Hence $z=\frac{1+\sqrt[10]{(-1)}}{\sqrt[10]{(-1)} -1}$.
