\documentclass[12pt]{book}
\usepackage{pas-exos}
%\usepackage{shortlst,amsmath}
\begin{document}
\chapter*{Expressions litt�rales}

\exercice % 1

R�duire au maximum les expressions suivantes :

\begin{shortenumerate}
\item $A=5x+3x^2+7x+5x^2+3$
\item $B=13a^2+12a-3a-2a^2$
\item $C=8xy+x^2+12xy+3x^2-4xy-x^2$
\end{shortenumerate}

\exercice % 2

D�velopper les expressions suivantes en r�duisant le r�sultat au maximum :

\begin{shortenumerate}
\item $A=3(x+4)$
\item $B=2(3x-4)$
\item $C=5(2-3x)$
\item $D=6(x+3)+4(3-x)$
\item $E=2(3x+1)+3(2x+1)$
\end{shortenumerate}

\exercice % 3

Factoriser au maximum les expressions suivantes :

\begin{shortenumerate}
\item $A=35x+25$
\item $B=16a+2a^2$
\item $C=4x-2x^2$
\item $D=20a-4b$
\item $E=5b-ab$
\end{shortenumerate}

\exercice % 4

\begin{minipage}{.4\textwidth}
\begin{center}
\begin{tikzpicture}[>=stealth']
\draw (0,0) rectangle +(5,-2);
\draw (4,-2) rectangle +(1,-1);
\draw[<->] (5.5,0) -- (5.5,-2) node[fill=white,midway] {$2x$};
\draw[<->] (5.5,-2) -- (5.5,-3) node[fill=white,midway] {$x$};
\draw[<->] (0,.5) -- (5,.5) node[fill=white,midway] {$7$};
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{.6\textwidth}
\begin{enumerate}
\item Montrer que le p�rim�tre de la figure ci-contre est �gal � $\mathcal{P}=6x+14$.
\item Montrer que l'aire de la figure ci-contre est �gale � $\mathcal{A}=x(14+x)$.
\end{enumerate}
\end{minipage}

\correction

\exercice % 1

\begin{minipage}{.5\textwidth}
$A=5x+3x^2+7x+5x^2+3$\\
$A=5x+7x+3x^2+5x^2+3$\\
\underline{$A=12x+8x^2+3$}
\end{minipage}
\begin{minipage}{.5\textwidth}
$B=13a^2+12a-3a-2a^2$\\
$B=13a^2-2a^2+12a-3a$\\
\underline{$B=11a^2+9a$}
\end{minipage}

\bigskip

$C=8xy+x^2+12xy+3x^2-4xy-x^2$\\
$C=8xy+12xy-4xy+x^2+3x^2-x^2$\\
$C=20xy-4xy+4x^2-x^2$\\
\underline{$C=16xy+3x^2$}

\exercice % 2

\begin{minipage}{.33\textwidth}
$A=3(x+4)$\\
$A=3\times x + 3\times 4$\\
\underline{$A=3x+12$}
\end{minipage}
\begin{minipage}{.33\textwidth}
$B=2(3x-4)$\\
$B=2\times 3x -2\times 4$\\
\underline{$B=-6x-8$}
\end{minipage}
\begin{minipage}{0.33\textwidth}
$C=5(2-3x)$\\
$C=5\times 2 -5\times 3x$\\
\underline{$C=10-15x$}
\end{minipage}

\bigskip

\begin{minipage}{.5\textwidth}
$D=6(x+3)+4(3-x)$\\
$D=6x+18+12-4x$\\
$D=6x-4x+18+12$\\
\underline{$D=2x+30$}
\end{minipage}
\begin{minipage}{.5\textwidth}
$E=2(3x+1)+3(2x+1)$\\
$E=6x+2+6x+3$\\
$E=6x+6x+2+3$\\
\underline{$E=12x+5$}
\end{minipage}

\exercice % 3

\begin{minipage}{.33\textwidth}
$A=35x+25$\\
$A=\boxed{5}\times 7x+\boxed{5}\times 5$\\
\underline{$A=5(7x+5)$}
\end{minipage}
\begin{minipage}{.33\textwidth}
$B=16a+2a^2$\\
$B=\boxed{2a}\times 8 + \boxed{2a}\times a$\\
\underline{$B=2a(8+a)$}
\end{minipage}
\begin{minipage}{.33\textwidth}
$C=4x-2x^2$\\
$C=\boxed{2x}\times 2 - \boxed{2x}\times x$\\
\underline{$C=2x(2-x)$}
\end{minipage}

\bigskip

\begin{minipage}{.5\textwidth}
$D=20a-4b$\\
$D=\boxed{4}\times 5a-\boxed{4}\times b$\\
\underline{$D=4(5a-b)$}
\end{minipage}
\begin{minipage}{.5\textwidth}
$E=5b-ab$\\
$A=\boxed{b}\times 5-\boxed{b}\times a$\\
\underline{$E=b(5-a)$}
\end{minipage}

\exercice % 4

1. Le p�rim�tre de la figure est la longueur rouge suivante :

\medskip

\begin{minipage}{.5\textwidth}
\begin{center}
\begin{tikzpicture}[>=stealth']
\draw (0,0) rectangle +(5,-2);
\draw (4,-2) rectangle +(1,-1);
\draw[<->] (5.5,0) -- (5.5,-2) node[fill=white,midway] {$2x$};
\draw[<->] (5.5,-2) -- (5.5,-3) node[fill=white,midway] {$x$};
\draw[<->] (0,.5) -- (5,.5) node[fill=white,midway] {$7$};
\draw[<->] (4,-3.5) -- (5,-3.5) node[fill=white,midway] {$x$};
\draw[<->] (3.5,-2) -- (3.5,-3) node[fill=white,midway] {$x$};
\draw[<->] (0,-1.75) -- (4,-1.75) node[fill=white,midway] {$7-x$};
\draw[<->] (-.5,0) -- (-.5,-2) node[fill=white,midway] {$2x$};
\draw[very thick,red] (0,0) -- (5,0) -- (5,-3) -- (4,-3) -- (4,-2) -- (0,-2) -- cycle;
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{.5\textwidth}
$\mathcal{P}=7+2x+x+x+x+(7-x)+2x$\\
$\mathcal{P}=2x+x+x+x-x+2x+7+7$\\
\underline{$\mathcal{P}=6x+14$}
\end{minipage}

\bigskip

2. L'aire de la figure est �gale � la somme de l'aire du rectangle vert et celle du rectangle bleu :

\medskip

\begin{minipage}{.5\textwidth}
\begin{center}
\begin{tikzpicture}[>=stealth']
\filldraw[fill=green!20,draw=green!50!black] (0,0) rectangle +(5,-2);
\filldraw[fill=blue!20,draw=blue!50!black] (4,-2) rectangle +(1,-1);
\draw[<->] (5.5,0) -- (5.5,-2) node[fill=white,midway] {$2x$};
\draw[<->] (5.5,-2) -- (5.5,-3) node[fill=white,midway] {$x$};
\draw[<->] (0,.5) -- (5,.5) node[fill=white,midway] {$7$};
\fill[white] (0,-5.5) circle (1pt);
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{.5\textwidth}
L'aire du rectangle vert est :
\[ \mathcal{A}_{\textrm{vert}}=7\times 2x = 14x\]
L'aire du rectangle bleu est :
\[ \mathcal{A}_{\textrm{bleu}}=x\times x = x^2\]
Donc l'aire totale est :\\
$\mathcal{A}=\mathcal{A}_{\textrm{vert}}+\mathcal{A}_{\textrm{bleu}}$\\
$\mathcal{A}=14x+x^2$\\
$\mathcal{A}=\boxed{x}\times 14+\boxed{x}\times x$\\
\underline{$\mathcal{A}=x(14+x)$}
\end{minipage}
\end{document}