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\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1316554716284--
%%--Calculs de base sur les complexes--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
\begin{enumerate}
\item Pour tout nombre complexe $z$, on pose:%
\[
P(z)=z^{3}+(-2+3i)z^{2}+(13-i)z+(-6-10i).
\]


\'{E}crire sous forme alg\'{e}brique $P(i)$, $P(3)$ et $P(1+i);$

\item Donner la forme alg\'{e}brique de chacun des nombres complexes
propos\'{e}s:%
\[
z_{1}=\frac{4-6i}{3+2i}\quad;\quad z_{2}=\frac{3-6i}{3+i}+\frac{4}{1-3i}%
\quad;\quad z_{3}=\left(  \frac{4-6i}{2-3i}\right)  \left(  \frac{1+3i}%
{3+2i}\right)  ;
\]


\item Pour tout nombre complexe $z=x+iy\quad(x$ et $y$ r\'{e}els$),$ on pose
$f(z)=z^{2}-z.$

D\'{e}terminer la partie r\'{e}elle et la partie imaginaire de $f(z)$ en
fonction de $x$ et de $y.$
\end{enumerate}
\end{exercise}

\begin{exercise}
\'{E}crire, en fonction de $\overline{z},$ le conjugu\'{e} des complexes
suivants:%
\[
z_{1}=2+3iz\quad;\quad z_{2}=(1+iz)(1-2z)\quad;\quad z_{3}=\frac{z-i}%
{3+z}\quad;\quad z_{4}=\left(  1+2iz\right)  ^{4}.
\]

\end{exercise}

\begin{exercise}
R\'{e}soudre dans $%
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\mathbb{C}
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\begin{enumerate}
\item $\left\{
\begin{array}
[c]{c}%
2iz+z^{\prime}=2i\\
3z-iz^{\prime}=1
\end{array}
\right.  ;$

\item $z^{2}-2\left(  1+\sqrt{2}\right)  z+2(\sqrt{2}+2)=0;$

\item $z^{2}-2\left(  \cos\theta\right)  z+1=0,$ o\`{u} $\theta\in%
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\mathbb{R}
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.$
\end{enumerate}%

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\end{exercise}


\end{document}