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\begin{center}\section*{\textcolor{red}{Correction des exercices de factorisations}}\end{center}

\subsection{}
Factoriser les expressions suivantes : 

\begin{enumerate}[$A(x)=$ ]

\medskip

\item $27x-36=9\times 3x-9\times 4=\boxed{\textcolor{red}{9(3x-4)}}$

\medskip

\item $D=27x^4-18x^3-15x^2=3x^2\times 9x^2-3x^2\times 6x-3x^2\times 5=\boxed{\textcolor{red}{3x^2\left(9x^2-6x+5\right)}}$

\medskip

\item $x^5+x^4=x^4\times x+x^4\times 1=\boxed{\textcolor{red}{x^4(x+1)}}$

\medskip

\item $5(x+1)+x(x+1)=\boxed{\textcolor{red}{(x+1)(5+x)}}$

\medskip

\item $\textbf{\textcolor{red}{(x-1)}}(2x+3)+\textbf{\textcolor{red}{(x-1)}}(5x-2)=(x-1)\left[(2x+3)+(5x-2)\right]=(x-1)(2x+3+5x-2)=\boxed{\textcolor{red}{(x-1)(7x+1)}}$

\medskip

\item $7\textcolor{red}{\textbf{(x -7)}} -x(x -7)+4\textcolor{red}{\textbf{(x -7)}}=(x -7)(7 -x+4)=\boxed{\textcolor{red}{(x -7)(11 -x) }}$

\medskip

\item $(5x+7)\textbf{\textcolor{red}{(x -1)}}+\textbf{\textcolor{red}{(x -1)}}(3x -4)=(x -1)[(5x+7)+(3x -4)]=(x -1)(5x+7+3x -4)=\boxed{\textcolor{red}{(x -1)(8x+3)}}$

\medskip

\item $(3x -2)(x -5)+(x -5)^2=(3x -2)\textbf{\textcolor{red}{(x -5)}}+\textbf{\textcolor{red}{(x -5)}}(x -5)=(x -5)[(3x -2)+(x -5)]=\boxed{\textcolor{red}{(x -5)(4x -7)}}$

\medskip

\item $(x+7)(5x+2) -3(5x+2)2=(x+7)\textbf{\textcolor{red}{(5x+2)}} -3\textbf{\textcolor{red}{(5x+2)}}(5x+2)=(5x+2)[(x+7) -3(5x+2)]\\
=(5x+2)(x+7 -15x -6)=\boxed{\textcolor{red}{(5x+2)( -14x+1)}}$

\end{enumerate}



\subsection{\textcolor{blue}{(Un peu plus subtil)}}
Factoriser les expressions suivantes :

\begin{enumerate}[$A(x)=$]
\item $(5x -2)+4(2x+1)(5x -2)=\textbf{\textcolor{red}{(5x -2)}}\times 1+4(2x+1)\textbf{\textcolor{red}{(5x -2)}}=(5x -2)[1+4(2x+1)]=(5x -2)(1+8x+4)\\
=\boxed{\textcolor{red}{(5x -2)(5+8x)}}$	

\medskip

\item $7x(2x+3)+2x+3=7x(2x+3)+(2x+3) 1=\boxed{\textcolor{red}{(2x+3)(7x+1)}}$  	   								

\medskip

\item $(3x+5)(x -1)+(x -1)=(3x+5)(x -1)+(x -1) 1=(x -1)[(3x+5)+1]=(x -1)(3x+5+1)	=\boxed{\textcolor{red}{(x -1)(3x+6)}}$ \textbf{\textcolor{blue}{(forme attendue)}}\\
$=(x -1)\times 3(x+2)=\boxed{\textcolor{red}{3(x -1)(x+2)}}$ \textbf{\textcolor{blue}{forme améliorée}} .  

\medskip

\item $(7x -2)(3x+4) -(3x+4)	=(7x -2)(3x+4) -(3x+4) 1=(3x+4)[(7x -2) -1]=(3x+4)(7x -2 -1)\\
=\boxed{\textcolor{red}{(3x+4)(7x -3)}}$	


\medskip

\item $(5x -1)(2x+3) -5x+1=(5x -1)(2x+3) -(5x -1)=(5x -1)(2x+3) -(5x -1) 1=(5x -1)[(2x+3) -1]\\
=(5x -1)(2x+3 -1)=\boxed{\textcolor{red}{(5x -1)(2x+2)}}$ \textcolor{blue}{(\textbf{forme attendue})}\\
$=\boxed{\textcolor{red}{2(5x -1)(x+1)}}$ \textcolor{blue}{(\textbf{forme améliorée})}    
\end{enumerate}
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\end{document}  