\documentclass[12pt, a4paper]{exam}%answers,addpoints, answers,
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\usepackage[french]{babel}
\usepackage{fontspec}
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\usepackage[ddmmyyyy]{datetime}
\usepackage{xparse} % Required for advanced argument parsing

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% Exam settings
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\everymath{\displaystyle}
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\newenvironment{mycontent}{%
\noindent
\borders\\[1.5cm]
\noindent
\exo{Exercice 1: }\\
Soit le polynome ~$P(x)=x^3-\sqrt2x^2-9x+9\sqrt2$~.
\begin{questions}
\question
  Montrer que ~$\sqrt2$~ est une racine du polynome ~$P(x)$~
\question
  Déterminer le polynome ~$Q(x)$~ tel que ~$P(x)=(x-\sqrt2)Q(x)$~
\question
  Montrer que ~$Q(x)$~ est divisible par 3
\question
  Ecrire ~$P(x)$~ sous la forme d'un produit de binomes
\question
  Résoudre l'équation ~$P(x)=0$~ et l'inéquation ~$P(x)<0$~
\end{questions}
\sticker[5.5cm][0]{(15,1)}{%
\textit{<<Mathematics is not about numbers, equations, or algorithms: it is about understanding.>>}\\
\textbf{— William Paul Thurston}
}
\exo{Exercice 2: }
\begin{questions}
\question
  Résoudre dans ~$\mathbb{R}$~ : ~$|2x-6|+|x+6|=5$~
\question
  Soit le polynome ~$P(x)=(x-3)(2x^2-3x+1)$~
  \begin{parts}
  \part
  Vérifier que 1  est une racine du polynome ~$2x^2-3x+1$~
  \part
  Effectuer la division euclidienne de ~$2x^2-3x+1$~ par ~$x-1$~
  \part
  Factoriser ~$2x^2-3x+1$~ sous forme de produit de binomes
  \part
    Résoudre ~$2x^2-3x+1=0$~
  \end{parts}
\question
  Résoudre l'équation ~$P(x)=0$~ et l'inéquation ~$P(x)\ge0$~
\end{questions}
\stamp{15}{2}
\exo{Exercice 3: }\\
	Soit le polynome \(P(x)=x^3-(a-b)x^2+(a-3b-1)x+2\sqrt{2}\)
\begin{questions}
  \question
	 Determiner les nombres \(a\) et \(b\) pour que \(P(x)\) soit divisible par \(x-2\) et \(x+\sqrt{2}\)
  \question
	 On pose \(a=3\) et \(b=\sqrt{2}\)
   \begin{parts}
     \part
       Déterminez un polynome \(Q(x)\) tel que \(P(x)=(x-2)Q(x)\)
     \part
        Calculer \(Q(-\sqrt{2}\) puis factoriser \(P(x)\)
    \part
       Résoudre \(x\in \mathbb{R}\quad P(x) < 0\)
   \end{parts}
   \question
	 On suppose que \(x\in]0,1[\). Montrer que \(\sqrt{2}\) est une
	approximation de \(P(x)\) à la précision \(1+\sqrt{2}\)
\end{questions}
\exo{Exercice 4: }\\
Soient les polynomes:\\ ~$A(x)= \sqrt2x^2-3x-5\sqrt2$~ et ~$B(x)=  x^2-2x+4$~ et
~$C(x)= x^2-6x+9$~
\begin{questions}
 \question
  Résoudre dans ~$\mathbb{R}$~ les équations ~$A(x)=0$~,  ~$B(x)=0$~ et  ~$C(x)=0$~
 \question
  Factoriser les polynomes ~$A(x)$~, ~$B(x)$~ et ~$C(x)$~
 \question
  Donner les tableaux de signes des polynomes ~$A(x)$~, ~$B(x)$~ et ~$C(x)$~
 \question
  Résoudre dans ~$\mathbb{R}$~ les inéquations ~$A(x)\le0$~,  ~$B(x)\ge0$~ et  ~$C(x)<0$~
\end{questions}
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    \begin{mycontent}\end{mycontent}
  \else
      %
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	\begin{mycontent}\end{mycontent}
 %   \newpage
 % \begin{mycontent}\end{mycontent}
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