Question

$$I + P ( x ) = x ^ { 3 } - 1 , P ( 1 ) + P ( - 1 )$$

Solve for x

$\left\{\begin{matrix}x=-\frac{6^{\frac{2}{3}}\left(1+\sqrt{3}i\right)\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)^{-\frac{1}{3}}\left(-\left(1+\sqrt{3}i\right)\times \left(3\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)\right)^{\frac{2}{3}}+6\times 2^{\frac{2}{3}}P\right)}{72}\text{; }x=\frac{18^{\frac{2}{3}}\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)^{-\frac{1}{3}}\left(\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)^{\frac{2}{3}}+\sqrt[3]{3}\times 2^{\frac{2}{3}}P\right)}{18}\text{; }x=-\frac{6^{\frac{2}{3}}\left(-\sqrt{3}i+1\right)\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)^{-\frac{1}{3}}\left(-\left(-\sqrt{3}i+1\right)\times \left(3\left(\sqrt{3\left(27I^{2}+54I-4P^{3}+27\right)}+9I+9\right)\right)^{\frac{2}{3}}+6\times 2^{\frac{2}{3}}P\right)}{72}\text{, }&P\neq 0\text{ and }a=0\\x=\frac{\left(-1+\sqrt{3}i\right)\sqrt[3]{I+1}}{2}\text{; }x=\sqrt[3]{I+1}\text{; }x=-\frac{\left(1+\sqrt{3}i\right)\sqrt[3]{I+1}}{2}\text{, }&P=0\text{ and }a=0\end{matrix}\right.$