Source Latex: Cours de mathématiques en Première STI2D


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Type: Cours
File type: Latex, tex (source)
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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
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Source Latex du cours de mathématiques

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    pdfauthor={Yoann Morel},
    pdfsubject={Cours math�matiques 1�re STI2D: synth�se sur le second
    degr�},
    pdftitle={Trin�me du second degr� - synth�se},
    pdfkeywords={Math�matiques, 1STI, premi�re, STI, STI2D, 
      second degr�, 2nd degr�, polyn�me
    }
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vspace*{-1cm}

%\ul{Nom:}
\hspace{5cm} 
{\Large Trin�me du second degr� : synth�se}
\hfill $1^{\text{�re}}\,$ STI2D
\vspace{0.8cm}

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

\vspq
Le discriminant du trin�me 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 r�elles 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 repr�sentative de $f$, 
  si $a>0$

  \vspace{1.2cm}\ \, 
  &

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  &
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  &
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    \put(1.8,.5){$\frac{-b}{2a}$}
  \end{pspicture}	
  \\\hline

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

  \vspace{1cm}\ \, 
  &

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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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