The complex plane is equipped with a direct orthonormal coordinate system. Consider the equation $$( E ) : \quad z ^ { 4 } + 2 z ^ { 3 } - z - 2 = 0$$ with unknown complex number $z$.
Give an integer solution of ( $E$ ).
Prove that, for every complex number $z$, $$z ^ { 4 } + 2 z ^ { 3 } - z - 2 = \left( z ^ { 2 } + z - 2 \right) \left( z ^ { 2 } + z + 1 \right) .$$
Solve equation ( $E$ ) in the set of complex numbers.
The solutions of equation ( $E$ ) are the affixes of four points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ in the complex plane such that ABCD is a non-crossed quadrilateral. Is quadrilateral ABCD a rhombus? Justify.
The complex plane is equipped with a direct orthonormal coordinate system. Consider the equation
$$( E ) : \quad z ^ { 4 } + 2 z ^ { 3 } - z - 2 = 0$$
with unknown complex number $z$.
\begin{enumerate}
\item Give an integer solution of ( $E$ ).
\item Prove that, for every complex number $z$,
$$z ^ { 4 } + 2 z ^ { 3 } - z - 2 = \left( z ^ { 2 } + z - 2 \right) \left( z ^ { 2 } + z + 1 \right) .$$
\item Solve equation ( $E$ ) in the set of complex numbers.
\item The solutions of equation ( $E$ ) are the affixes of four points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ in the complex plane such that ABCD is a non-crossed quadrilateral.\\
Is quadrilateral ABCD a rhombus? Justify.
\end{enumerate}