%\usepackage{pst-plot}

\documentclass[a4paper]{article}%
\usepackage{amsfonts}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage{amsmath,amssymb}
\usepackage[a4paper]{geometry}
\usepackage{xspace}
\usepackage{fancyhdr}
\usepackage{enumerate}
\usepackage{multicol}
\usepackage{theorem}
\usepackage{graphics}
\usepackage{graphicx}
\usepackage{lastpage}
\usepackage[french]{layout}
\usepackage{boxedminipage}
\usepackage{fancybox}
\usepackage{sectsty}
\usepackage{afterpage}
\usepackage[francais]{babel}%
\usepackage{amsmath}%
\setcounter{MaxMatrixCols}{30}%
\usepackage{amssymb}
%TCIDATA{OutputFilter=latex2.dll}
%TCIDATA{Version=5.00.0.2552}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{LastRevised=Sunday, August 29, 2004 20:39:07}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
\geometry{a4paper,marginpar=2cm,lmargin=0cm,rmargin=2cm,marginparsep=0pt,includemp=true,reversemp=true,headsep=-0.2in}
\setlength{\theorempostskipamount}{2\baselineskip}
\theoremheaderfont{\large\bfseries}
{\theorembodyfont{\large\upshape\mdseries}\theoremstyle{break}
\newtheorem{theorem}{Théorème}
}
\newcounter{Cack}[section]
{\theorembodyfont{\large\upshape}\theoremstyle{break}
\newtheorem{acknowledgement}[Cack]{A savoir}
}
\newcounter{Calg}[section]
{\theorembodyfont{\upshape}\theoremstyle{break}
\newtheorem{algorithm}[Calg]{Méthode}
}
\newcounter{Caxi}[section]
{\theorembodyfont{\itshape}
\newtheorem{axiom}[Caxi]{Axiome}
}
\newcounter{Ccas}[section]
{\theorembodyfont{\upshape}
\newtheorem{case}[Ccas]{Cas}
}
\newcounter{Ccorr}[section]
\renewcommand{\theCcorr}{\Roman{Ccorr}}
{\theorembodyfont{\upshape}\theoremstyle{break}
\newtheorem{claim}[Ccorr]{Correction de l'exercice}
}
\newcounter{Cconc}[theorem]
{\theorembodyfont{\large\mdseries}
\newtheorem{conclusion}[Cconc]{Conclusion}
}
\newcounter{Ccond}[theorem]
{\theorembodyfont{\large\mdseries}
\newtheorem{condition}[Ccond]{Hypothèse}
}
\newcounter{Cconj}[section]
{\theorembodyfont{\upshape}
\newtheorem{conjecture}[Cconj]{Conjecture}
}
\newcounter{Ccor}[section]
{\theorembodyfont{\upshape}
\newtheorem{corollary}[Ccor]{Corollaire}
}
\newcounter{Ccrit}[section]
{\theorembodyfont{\upshape}
\newtheorem{criterion}[Ccrit]{Critère}
}
\newcounter{Cdef}[section]
\newcounter{Cnot}[section]
{\theorembodyfont{\large\upshape}\theoremstyle{break}
\newtheorem{definition}[Cdef]{Définition}
\newtheorem{notation}[Cnot]{Notation}
}
\newcounter{Cexa}[section]
{\theorembodyfont{\slshape}
\newtheorem{example}[Cexa]{Exemple}
}
\newcounter{Cexe}[section]
\renewcommand{\theCexe}{\Roman{Cexe}}
{\theorembodyfont{\upshape}\theoremstyle{break}
\newtheorem{exercise}[Cexe]{Exercice}
}
\newcounter{Clem}[section]
{\theorembodyfont{\upshape}
\newtheorem{lemma}[Clem]{Lemme}
}
\newcounter{Cprob}[section]
\renewcommand{\theCprob}{\Roman{Cprob}.}
{\theorembodyfont{\upshape}\theoremstyle{break}
\newtheorem{problem}[Cprob]{Problème}
}
{\theorembodyfont{\large\mdseries}\newcounter{Cprop}[section]\theoremstyle{break}
\newtheorem{proposition}[Cprop]{Propriété}
}
\newcounter{Crem}[section]
{\theorembodyfont{\itshape}
\newtheorem{remark}[Crem]{Remarque}
}
\newcounter{Csol}[section]
{\theorembodyfont{\upshape}\theoremstyle{break}
\newtheorem{solution}[Csol]{Solution}
}
\newcounter{Csum}[section]{\theorembodyfont{\upshape}
\newtheorem{summary}[Csum]{En résumé}
}
\newenvironment{proof}[1][Démonstration]{\noindent\textbf{#1:}}{\ \fbox{cqfd.}}
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand{\theenumii}{\alph{enumii}}
\renewcommand{\theenumiii}{\roman{enumiii}}
\renewcommand{\labelenumi}{\textbf{\theenumi)}}
\renewcommand{\labelenumii}{\textbf{\theenumii)}}
\renewcommand{\labelenumiii}{\textbf{\theenumiii.}}
\begin{document}

%%%%%%%%%%b{gedeonInfo}%%%%%%%%%
%%--Document N° 1316448625641--
%%--Factorisation de polynômes et fonctions rationnelles--
%%%%%%%%%%e{gedeonInfo}%%%%%%%%%


\begin{exercise}
On considère le polynôme $P$ défini sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ par: $P(x)=x^{3}+x^{2}-4x-4$.

\begin{enumerate}
\item Calculer $P(-1)$ et factoriser $P(x);$

\item En utilisant l'expression factorisée de $P(x),$ résoudre l'équation:
$P(x)=0;$

\item En utilisant l'expression factorisée de $P(x),$ résoudre l'inéquation:
$P(x)\geq0.$\medskip
\end{enumerate}
\end{exercise}

\begin{exercise}
$\vspace*{-0.35cm}$

\begin{enumerate}
\item Déterminer $a,$ $b$ et $c$ tels que:%
\[
\forall x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash\{-1\},\quad\frac{2x^{2}+5x+7}{x+1}=ax+b+\frac{c}{x+1};
\]


\item Déterminer $a$ et $b$ tels que:%
\[
\forall x\in%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\backslash\{1;-1\},\quad\frac{2}{x^{2}-1}=\frac{a}{x-1}+\frac{b}{x+1}.
\]
\medskip
\end{enumerate}
\end{exercise}

\begin{exercise}
Étant donné un réel $a$ fixé, on considère le polynôme $P$ défini sur $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ par: $P(x)=ax^{3}+4x^{2}-28x+16.$

\begin{enumerate}
\item
\begin{enumerate}
\item Déterminer la valeur de $a$ pour que l'on ait $P(2)=0;$

\item $a$ ayant la valeur trouvée au 1) a), factoriser $P(x).$
\end{enumerate}

\item
\begin{enumerate}
\item Déterminer la valeur de $a$ pour que le point $M$ de coordonnées
$(1;-8)$ appartienne à la courbe représentative de $P;$

\item $a$ ayant la valeur trouvée au 2) a), donner le degré de $P.$
\end{enumerate}
\end{enumerate}
\end{exercise}


\end{document}