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\author{steven.jeanneret}
\pdfinfo{
  /Title (maths-1b-1-bases-d-algebre.pdf)
  /Creator (Cheatography)
  /Author (steven.jeanneret)
  /Subject (Maths\_1B\_1\_Bases\_d'Algebre Cheat Sheet)
}

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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Maths\_1B\_1\_Bases\_d'Algebre Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{steven.jeanneret} via \textcolor{DarkBackground}{\uline{cheatography.com/32278/cs/9916/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}steven.jeanneret \\
  \uline{cheatography.com/steven-jeanneret} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 21st November, 2016.\\
   Updated 21st November, 2016.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
  www.readability-score.com
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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{Les intervalles}}  \tn
% Row 0
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XƐ{[}a,b{]} & a \textless{}= x \textless{}= b & intervalle fermé \tn 
% Row Count 2 (+ 2)
% Row 1
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XƐ{]}a,b{[} & a \textless{} x \textless{} b & intervalle ouvert \tn 
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XƐ{[}a,b{[} & a \textless{}= x \textless{} b & intervalle ouvert à droite \tn 
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\mymulticolumn{1}{x{8.4cm}}{-5 \textless{} 6 | + 2} \tn 
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\mymulticolumn{1}{x{8.4cm}}{} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Le signe de l'inégalité n'est pas changé.} \tn 
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\begin{tabularx}{8.4cm}{X}
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\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Si on multiplie par un nombre négatif on inverse le sens de l'inégalite} \tn 
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\mymulticolumn{1}{x{8.4cm}}{} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Il y a inversion de signe si l'inégalité de départ est fausse en valeur absolue.} \tn 
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\mymulticolumn{1}{x{8.4cm}}{} \tn 
\mymulticolumn{1}{x{8.4cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Le signe de l'inégalité n'est pas changé. On prend seulement les nombres positifs.} \tn 
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| a + b | & si a + b = 0 & 0 \tn 
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% Row 4
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% Row 5
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% Row 6
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% Row 7
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% Row 8
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a\textasciicircum{}n\textasciicircum{} - b\textasciicircum{}n\textasciicircum{} & (a + b) (a - b) (...) \tn 
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% Row 10
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(...) & S'obtient par division polynomiale \tn 
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% Row 1
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a\textasciicircum{}2\textasciicircum{} + 2a & a(2 + a) \tn 
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% Row 2
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\mymulticolumn{2}{x{8.4cm}}{Reconnaissance d'identité remarquable :} \tn 
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% Row 3
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x\textasciicircum{}2\textasciicircum{} - 9 & (x + 3) (x - 3) \tn 
% Row Count 4 (+ 1)
% Row 4
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\mymulticolumn{2}{x{8.4cm}}{Regroupement :} \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
aX\textasciicircum{}2\textasciicircum{} + bX + c & a( X\textasciicircum{}2\textasciicircum{} + bX{\bf{/a}} + c{\bf{/a}}) \tn 
% Row Count 7 (+ 2)
% Row 6
\SetRowColor{LightBackground}
a( X\textasciicircum{}2\textasciicircum{} + {\bf{2}}bX/{\bf{2}}a + c/a) & a ((X + {\bf{b/2a}})\textasciicircum{}2\textasciicircum{}-b\textasciicircum{}2\textasciicircum{}/4a\textasciicircum{}2\textasciicircum{}+c/a) \tn 
% Row Count 9 (+ 2)
% Row 7
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{a((X+ b/2a)\textasciicircum{}2\textasciicircum{} - (b\textasciicircum{}2\textasciicircum{}+4ac)/4a} \tn 
% Row Count 10 (+ 1)
% Row 8
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x1 = (-b + √b\textasciicircum{}2\textasciicircum{} - 4ac )/ 2a & x2 = (-b - √b\textasciicircum{}2\textasciicircum{} - 4ac )/ 2a \tn 
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Fonctions quadratiques :}}  \tn
% Row 0
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f1(x) = x\textasciicircum{}2\textasciicircum{} & Donne une fonction de type U \tn 
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% Row 1
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f2(x) = (x - q)\textasciicircum{}2\textasciicircum{} & Donne une fonction de type U avec un décalage q (en x) \tn 
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f3(x) = p(x-q)\textasciicircum{}2\textasciicircum{} & Plus p est grand plus la fonction est serré Si p est négatif la fonction part contre le bas. \tn 
% Row Count 9 (+ 4)
% Row 3
\SetRowColor{white}
f4(x) = p (x-q)\textasciicircum{}2\textasciicircum{} + r & Ajouter r décale la fonciton verticalement (axe y) \tn 
% Row Count 12 (+ 3)
% Row 4
\SetRowColor{LightBackground}
y = a ( x - b)\textasciicircum{}2\textasciicircum{}+ o & Le point S est (b; o) \tn 
% Row Count 14 (+ 2)
% Row 5
\SetRowColor{white}
S est un maximum si a \textless{} 0 & S est un minimum si a \textgreater{} 0 \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Fonctions quadratiques (les racines)}}  \tn
% Row 0
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Les racines sont les intersections de la fonction avec l'axe x & x1 = 0 \& x2 = 0 \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
p(x-q)\textasciicircum{}2\textasciicircum{} + r = 0 & x-q = +ou- √-r/p \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
x1 = √(-r/p) + q & x2 = -√(-r/p) + q \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

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\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Forme Polynomiale en canonique}}  \tn
% Row 0
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\mymulticolumn{1}{x{8.4cm}}{Pour trouver une racine il faut la forme canonique:} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{f(x) = ax\textasciicircum{}2\textasciicircum{} + bx + c} \tn 
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% Row 2
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\mymulticolumn{1}{x{8.4cm}}{a{[}x\textasciicircum{}2\textasciicircum{}+2(b/2a)*x+(b/2a)\textasciicircum{}2\textasciicircum{}-(b/2a)\textasciicircum{}2\textasciicircum{}+c/a{]}} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{a(x + b/2a)\textasciicircum{}2\textasciicircum{} - (b\textasciicircum{}2\textasciicircum{} - 4ac)/4a} \tn 
% Row Count 5 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{On veut p(x - q)\textasciicircum{}2\textasciicircum{} + r} \tn 
% Row Count 6 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{r = - (b\textasciicircum{}2\textasciicircum{}-4ac)/4a} \tn 
% Row Count 7 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{q = -b/2a} \tn 
% Row Count 8 (+ 1)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{p = a} \tn 
% Row Count 9 (+ 1)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{On a donc : a (x - (-b/2a))\textasciicircum{}2\textasciicircum{} + ((b\textasciicircum{}2\textasciicircum{}-4ac)/4a)} \tn 
% Row Count 10 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.68 cm} x{4.32 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Factorisation de fonction}}  \tn
% Row 0
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p(x) = a(X - X1) (X - X2) & a(X\textasciicircum{}2\textasciicircum{}-X {\emph{ X1 - X }} X2 + X1 * X2) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
aX\textasciicircum{}2\textasciicircum{} - a(X1 + X2) X + a {\emph{ X1 }} X2 & On retrouve une équation du second degré. \tn 
% Row Count 5 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{X1 et X2 sont les racines du polynôme :} \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
P(X1) = a( X1 - X1) ( X1 - X2) = 0 & On sait que X1 - X1 = 0 \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
P(X2) = a( X2 - X1) ( X2 - X2) = 0 & On sait que X2 - X2 = 0 \tn 
% Row Count 10 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{Il est facile et toujours possible de passer de la forme factorisé à la forme polynômiale.} \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{S'il n'y a pas de racines réeles (intersection avec l'axe x) on ne pourra pas mettre le polynôme sous forme factorisé c'est donc un pôlynome irréductible!} \tn 
% Row Count 16 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
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\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Factorisation de fonction degré n}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{Pour passer de la forme factorisée à la forme pôlynomiale on procède comme une fonction de degré 2} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{Pour l'opération inverse il faut utiliser le Théorème Fondamental de l'algébre.} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{Tout polynôme Pn(x) de degré {\bf{n}} peut s'écrire comme le produit de k polynômes du premier degré et m polynômes irréductibles du seconde degré.} \tn 
% Row Count 9 (+ 4)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{k correspond au nombre de racines réelles de Pn(x)} \tn 
% Row Count 11 (+ 2)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{n = k + 2m} \tn 
% Row Count 12 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{m = (n - k)/2} \tn 
% Row Count 13 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Décomposition en fractions simples}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{f(x) = N(x) / D(x)} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{Si le degré du numérateur est supérieur ou égal au degré du dénominateur on effectue une division polynômiale et on pourra factoriser le reste.} \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{E1. Si nécessaire effectuer la division euclidienne et prendre {\bf{uniquement}} le reste pour les prochaines étapes :} \tn 
% Row Count 7 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{E2. Factoriser Dn(x) en produit soit de facteur linéaires (px + q)\textasciicircum{}n\textasciicircum{} soit/et en facteurs quadratiques irréductibles (ax\textasciicircum{}2\textasciicircum{} + bx + c)\textasciicircum{}m\textasciicircum{}} \tn 
% Row Count 10 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{E3. Pour chaque facteur (px + q)\textasciicircum{}n\textasciicircum{} écrire la somme de fractions : A1/(px+q) + A2/(px+q)\textasciicircum{}2\textasciicircum{} + An/(px+q)\textasciicircum{}n\textasciicircum{}} \tn 
% Row Count 13 (+ 3)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{E4. Pour chaque facteur (ax\textasciicircum{}2\textasciicircum{} +bx +c)\textasciicircum{}m\textasciicircum{} écrire la somme de fractions simples : ((B1*x + c1)/(ax\textasciicircum{}2\textasciicircum{}+bx+c)) + ((B2x + c2)/(ax\textasciicircum{}2\textasciicircum{}+bx+c)\textasciicircum{}2\textasciicircum{})+((Bmx + cm)/(ax\textasciicircum{}2\textasciicircum{}+bx+c)\textasciicircum{}m\textasciicircum{})} \tn 
% Row Count 17 (+ 4)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{E5. Calculer les constantes Ai, Bi, Cien posant que la fraction rationnelle Nk(x) / Dn(x) doit être {\bf{IDENTIQUE}} à sa décomposition en fractions simples.} \tn 
% Row Count 21 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{La décomposition en fractions simples concerne les fonctions rationelles irréductibles.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}