\documentclass[a4paper,10pt,french]{article}


\input macro_dwicky_final.tex
\geometry{verbose,letterpaper,tmargin=1cm,bmargin=1cm,lmargin=2cm,rmargin=2cm}

\begin{document}
\pagestyle{empty}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
%\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{Correction de l'interrogation n$^{\circ}1$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}
\setcounter{section}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro


\Exoi{\hfill (2 points)
\begin{multicols}{2}
\begin{enumerate}
\item $(a+b)^2=a^2+2ab+b^2$
\item $a^2-b^2=(a-b)(a+b)$
\end{enumerate}
\end{multicols}
}
\Exoi{\hfill (2 points) \\
\begin{enumerate}
\item $(x+y)^2-(x-y)^2=x^2+2xy+y^2-(x^2-2xy+y^2)=4xy$
\item Par conséquent $10002^2-9998^2=(10000+2)^2-(10000-2)^2=4\times 10000\times 2=80000$
\end{enumerate}}
\Exoi{\hfill (6 points)\\
\begin{enumerate}
\item 
\begin{enumerate}
\item $A=(x-1)^2(5-2x)=(x^2-2x+1)(5-2x)=5x^2-2x^3-10x+4x^2+5-2x=-2x^3+9x^2-12x+5$
\item $B=(-2(x-1))^2=(-2x+2)^2=4x^2-8x+4$
\item $C=(\sqrt{3}x+4)^2=3x-8 \sqrt{3}x+16 $
\end{enumerate}
\item 
\begin{enumerate}
\item $D=(4x-1)(3x-1)+(4x-1)=(4x-1)(3x-1+1)=(4x-1)(3x)=3x(4x-1)$
\item $E=(1-6x)^2-(1-6x)(2+5x)+4(1-6x)=(1-6x)(1-6x-2-5x+4)=(1-6x)(-11x+3)$
\item $F=9x^2-18x+9=(3x-3)^2$
\end{enumerate}
\end{enumerate}
}



\vfill

\setcounter{Exi}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	Cadre jaune
%\textbf{Nom : }$\dots\dots\dots\dots\dots$ \hfill \textbf{Pr\'enom : }$\dots\dots\dots\dots\dots$  \hfill Classe :$\dots\dots$\\
\psframebox[linestyle=solid,shadow=true,framearc=0.5,fillcolor=vert!30]{
\begin{minipage}{17cm}
\begin{center}
\vspace{1mm}
\textsc{\textbf{Correction de l'interrogation n$^{\circ}1$ }}
\end{center}
\vspace{-3mm}
\end{minipage}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%	 Interro bis


\Exoi{\hfill (2 points)
\begin{multicols}{2}
\begin{enumerate}
\item $a^2-2ab+b^2=(a-b)^2$
\item $(a+b)(a-b)=a^2-b^2$
\end{enumerate}
\end{multicols}
}
\Exoi{\hfill (2 points) \\
\begin{enumerate}
\item $(x+y)^2-(x-y)^2=x^2+2xy+y^2-(x^2-2xy+y^2)=4xy$
\item Par conséquent $10003^2-9997^2=(10000+3)^2-(10000-3)^2=4\times 10000\times 3=120000$
\end{enumerate}}

\Exoi{\hfill (6 points)\\
\begin{enumerate}
\item 
\begin{enumerate}
\item $A=(x-1)(5-2x)^2=(x-1)(25-20x+4x^2)=25x-20x^2+4x^3-25+20x-4x^2=4x^3-24x^2+45x-25$
\item $B=(-3(x+1))^2=(-3x-3)^2=9x^2-18x+9$
\item $C=(\sqrt{2}x+4)^2=2x+8 \sqrt{x}+16 $
\end{enumerate}
\item Factoriser les expressions suivantes :
\begin{enumerate}
\item $D=(2x-1)(3x-1)+2(2x-1)=(2x-1)(3x-1+2)=(2x-1)(3x+1)$
\item $E=(1-6x)^2-(1-6x)(2+5x)+(1-6x)=(1-6x)(1-6x-2-5x+1)=(1-6x)(-11x)=-11x(1-6x)$
\item $F=4x^2+12x+9=(2x+3)^2$
\end{enumerate}
\end{enumerate}
}

\end{document}
