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%\fancyhead[L]{Lycee Taghzirt\textbf{/}Prof MOSAID}
%\fancyhead[R]{TCSF-3\textbf{/}2h}
%\fancyhead[C]{\hspace*{2.5cm}2023-2024\textbf{/}Devoir 1 S02}
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\noindent\begin{tabularx}{\textwidth}{@{} lCr @{}}
    Lycee Taghzirt\textbf{/}Prof MOSAID &
    2023-2024\textbf{/}Devoir Libre 1 S02&
    TCSF-3\textbf{/}2h\\
    \bottomrule
\end{tabularx}
\mylink \hfill \mylink\\
\exe{1}(10pts)\\
\noindent
\begin{tabular}{@{}X{0.04\textwidth}|p{0.96\textwidth}}
    & \mylabel[green]{1} Résoudre dans \(\mathbb{R}\) l'équation:
        \(\frac{x}{\sqrt3}+\sqrt2=2\sqrt3x-1\) \\
    & \mylabel[green]{1} Résoudre dans \(\mathbb{R}\) l'équation: \(3x^2+\frac{1}{2}x-\sqrt2=0\) \\
    & \mylabel[green]{2} Résoudre dans \(\mathbb{R}\) l'équation: \(4x^2-2\sqrt2x+\frac{1}{4}=0\) \\
    & \mylabel[green]{3} En déduir les solutions de l'inéquation: \((3x^2+\frac{1}{2}x-\sqrt2)(4x^2-2\sqrt2x+\frac{1}{4})\ge0\) \\
    & \mylabel[green]{4} Résoudre dans \(\mathbb{R}\) l'équation: \(|3-x|+2|2x-1|-|2+3x|=1\) \\
    & \mylabel[green]{4} Résoudre dans \(\mathbb{R}\) l'équation: \(4x-2\sqrt{2x}+\frac{1}{4}=0\) \\
    & \mylabel[green]{4} Résoudre dans \(\mathbb{R}^2\) le système:
    \(
        \begin{cases}
            3x-2y=1\\
            x+4y=4
        \end{cases}
        \vspace*{-0.5cm}
    \)
\end{tabular}
\exe{2}(10pts)\\
\noindent
\begin{tabular}{@{}X{0.04\textwidth}|p{0.96\textwidth}}
    &Soit le polynome \(P(x)=x^3-(a-b)x^2+(a-3b-1)x+2\sqrt{2}\)\\
    &\mylabel[green]{1}. Determiner les nombres \(a\) et \(b\) pour que \(P(x)\) soit divisible par
        \(x-2\) et \(x+\sqrt{2}\)\\
    &\mylabel[green]{2} On pose \(a=3\) et \(b=\sqrt{2}\)\\
    &\mylabel[lightblue]{2.1} Déterminez un polynome \(Q(x)\) tel que \(P(x)=(x-2)Q(x)\)\\
    &\mylabel[lightblue]{2.2} Calculer \(Q(-\sqrt{2}\) puis factoriser \(P(x)\)\\
    &\mylabel[lightblue]{2.2} Résoudre \(x\in \mathbb{R}\quad P(x) < 0\)\\
    &\mylabel[green]{3} On suppose \(x\in]0,1[\). Montrer que \(\sqrt{2}\) est une
    approximation de \(P(x)\) à la précision \(1+\sqrt{2}\)\\
\end{tabular}
\\
\\
\textcolor{white}{.}\hfill \underline{MOSAID le \today}\\
\textcolor{white}{.}\hfill \mylink
\\
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