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Type: Cours
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Description
Synthèse de cours de mathématiques en 1ère STI2D - Second degré et polynômes
Niveau
Première STI2D
Mots clé
second degré, trinôme du second degré, polynôme, équation du second degré, synthèse de cours de mathématiques, maths, première, 1ère, STI2D
Voir aussi:

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    pdfsubject={Cours mathmatiques 1re STI2D: synthse sur le second
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    pdftitle={Trinme du second degr - synthse},
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      second degr, 2nd degr, polynme
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\vspace*{-1cm}

%\ul{Nom:}
\hspace{5cm} 
{\Large Trinme du second degr : synthse}
\hfill $1^{\text{re}}\,$ STI2D
\vspace{0.8cm}

Pour un trinme du second degr: 
\ul{$f(x)=ax^2+bx+c$}, o $a$, $b$ et $c$ sont trois rels, et $a\not=0$. 

\vspq
Le discriminant du trinme est \fbox{$\Delta=b^2-4ac$}.

\vspace{0.8cm}
\hspace{-1cm}
\begin{tabular}{|p{4cm}|p{4.8cm}|p{4.5cm}|p{4.5cm}|} \cline{2-4}
  \multicolumn{1}{c|}{}& \vspd\ct{$\Delta>0$} & \vspd\ct{$\Delta=0$} & \vspd\ct{$\Delta<0$} \\\hline

  \vspace{0.4cm}
  \begin{flushleft}
  Solution(s) de l'quation $f(x)=0$ 

  (racines de $f$) 
  \end{flushleft}
  %\vspace{0.cm}\ \,

  &2 solutions relles distinctes: 

  \[ x_1=\frac{-b-\sqrt{\Delta}}{2a}\]

  \[ x_2=\frac{-b+\sqrt{\Delta}}{2a}\]
  & une solution unique (double) : 

  \[\dsp x_0=\frac{-b}{2a}\]
  & \vspace{0.5cm}pas de solution \\\hline

  \rule[-1cm]{0.cm}{2cm}
  \hspace{-0.2cm}Factorisation de $f(x)$
  &$f(x)=a(x-x_1)(x-x_2)$ 
  &$f(x)=a(x-x_0)^2$
  &pas de factorisation \\\hline

  \vspace{-2cm}
  Courbe reprsentative de $f$, 
  si $a>0$

  \vspace{1.2cm}\ \, 
  &

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    \put(0.4,0.8){$x_1$}
    \put(3.3,0.8){$x_2$}
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  &
  \psset{xunit=1cm,yunit=1cm}
  \begin{pspicture}(0,0)(4,4)
    \psline[linewidth=0.8pt]{->}(0,1)(4.2,1)
    \psline[linewidth=0.8pt]{->}(1,-0.)(1,3.5)
    \psplot{0.5}{3.5}{x -2 add x -2 add mul 1 add}
    \put(1.8,0.5){$\frac{-b}{2a}$}
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  &
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    \put(1.8,.5){$\frac{-b}{2a}$}
  \end{pspicture}	
  \\\hline

  \vspace{-2cm}
  Courbe reprsentative de $f$, 
  si $a<0$

  \vspace{1cm}\ \, 
  &

  \psset{xunit=1cm,yunit=1cm}
  \begin{pspicture}(0,0)(4,4)
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  &
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  \\\hline

  \vspd  \vspd
  Signe de $f(x)$
  &
  \vspace{0.2cm}\hspace*{-0.3cm}
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    \!$x$\!\!    & $-\infty$              & $x_1$  &               &$x_2$& \hfill$+\infty$ \\\hline
    \!$f$\!\! &           Signe de $a$ & \raisebox{-0.2cm}[0.3cm][0.5cm]{\zb}    & Signe de -$a$ &\raisebox{-0.2cm}[0.3cm][0.5cm]{\zb}  &
    Signe de $a$  \\\hline
  \end{tabular}
  }\vspace{0.2cm}
  &
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  \begin{tabular}{|c|p{0.9cm}cp{0.9cm}|}\hline
    \!$x$\!\!    & $-\infty$              & $x_0$  & \hfill$+\infty$ \\\hline
    \!$f$\!\! &           Signe de $a$ &\raisebox{-0.2cm}[0.3cm][0.5cm]{\zb}    &    Signe de $a$  \\\hline
  \end{tabular}
  }\vspace{0.2cm}
  &
  \vspace{0.4cm}\hspace*{0.cm}
  \scalebox{0.8}{
  \begin{tabular}{|c|p{0.9cm}cr|}\hline
    \!$x$\!\! & $-\infty$ &            &$+\infty$ \\\hline
    \!$f$\!\! &           &Signe de $a$&  \\\hline
  \end{tabular}
  }\vspace{0.2cm}  \\\hline
\end{tabular}


\end{document}

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