137 lines
4.2 KiB
TeX
137 lines
4.2 KiB
TeX
\documentclass[a4paper,10pt]{article}
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\RequirePackage[utf8x]{inputenc}
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\RequirePackage[francais]{babel}
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\RequirePackage{amssymb}
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\RequirePackage{amsmath}
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\RequirePackage{amsfonts}
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\RequirePackage{subfig}
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\RequirePackage{graphicx}
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\RequirePackage{color}
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% Title Page
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\title{Calcul littéral et statistiques}
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\date{\today}
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\begin{document}
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\maketitle
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\section{Polynômes}
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\Block{set P = Polynom.random(["{a}", "{b}", "{c}"], ["{b}**2 - 4*{a}*{c} == 0"])}
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Résoudre l'équation suivante
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\begin{eqnarray*}
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\Var{P} & = & 0
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\end{eqnarray*}
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Solution:
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On commence par calculer le discriminant
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\Block{set Delta = Expression("{b}^2 - 4*{a}*{c}".format(a = P._coef[2], b = P._coef[1], c = P._coef[0]))}
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\begin{eqnarray*}
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\Delta & = & b^2-4ac \\
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\Var{Delta.simplify()|calculus(name="\\Delta")}
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\end{eqnarray*}
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\Block{set Delta = Delta.simplified()}
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\Block{if Delta > 0}
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Alors $\Delta = \Var{Delta} > 0$ donc il y a deux solutions
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\Block{set x1 = (-P._coef[1] - sqrt(Delta))/(2*P._coef[2])}
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\Block{set x2 = (-P._coef[1] + sqrt(Delta))/(2*P._coef[2])}
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\begin{eqnarray*}
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x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-P._coef[1]} - \sqrt{\Var{Delta}}}{2 \times \Var{P._coef[2]}} = \Var{x1 | round(2)} \\
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x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-P._coef[1]} + \sqrt{\Var{Delta}}}{2 \times \Var{P._coef[2]}} = \Var{x2 | round(2)}
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\end{eqnarray*}
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Les solutions sont donc $\mathcal{S} = \left\{ \Var{x1|round(2)}; \Var{x2|round(2)} \right\}$
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\Block{elif Delta == 0}
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Alors $\Delta = \Var{Delta} = 0$ donc il y a une solution
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\Block{set x1 = Expression("-{b}/(2*{a})".format(b = P._coef[1], a = P._coef[2]))}
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\begin{eqnarray*}
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x_1 = \frac{-b}{2a} = \Var{" = ".join(x1.simplify())}
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\end{eqnarray*}
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Les solutions sont donc $\mathcal{S} = \left\{ \Var{x1.simplified()}\right\}$
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\Block{else}
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Alors $\Delta = \Var{Delta} < 0$ donc il n'y a pas de solution.
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\Block{endif}
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\bigskip
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~\dotfill
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\bigskip
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\Block{set P = Polynom.random(["{a}", "{b}", "{c}"])}
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\Block{set Q = Polynom.random(["{a}", "{b}", "{c}"])}
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Résoudre l'équation suivante
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\begin{eqnarray*}
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\Var{P} & = & \Var{Q}
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\end{eqnarray*}
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Solution:
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On commence par se ramener à une équation de la forme $ax^2+bx+c = 0$.
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\begin{eqnarray*}
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\Var{P} = \Var{Q} & \Leftrightarrow & \Var{P} - (\Var{Q}) = 0 \\
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\Var{(P - Q)|calculus(name = "", sep = "\\Leftrightarrow", end = "= 0")}
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\end{eqnarray*}
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\Block{set R = (P-Q)[-1]}
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On cherche maintenant à résoudre l'équation $\Var{R} = 0$.
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On commence par calculer le discriminant
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\Block{set Delta = Expression("{b}^2 - 4*{a}*{c}".format(a = R._coef[2], b = R._coef[1], c = R._coef[0]))}
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\begin{eqnarray*}
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\Delta & = & b^2-4ac \\
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\Var{Delta.simplify()|calculus(name="\\Delta")}
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\end{eqnarray*}
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\Block{set Delta = Delta.simplified()}
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\Block{if Delta > 0}
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Alors $\Delta = \Var{Delta} > 0$ donc il y a deux solutions
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\Block{set x1 = (-R._coef[1] - sqrt(Delta))/(2*R._coef[2])}
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\Block{set x2 = (-R._coef[1] + sqrt(Delta))/(2*R._coef[2])}
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\begin{eqnarray*}
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x_1 & = & \frac{-b - \sqrt{\Delta}}{2a} = \frac{\Var{-R._coef[1]} - \sqrt{\Var{Delta}}}{2 \times \Var{R._coef[2]}} = \Var{x1 | round(2)} \\
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x_2 & = & \frac{-b + \sqrt{\Delta}}{2a} = \frac{\Var{-R._coef[1]} + \sqrt{\Var{Delta}}}{2 \times \Var{R._coef[2]}} = \Var{x2 | round(2)}
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\end{eqnarray*}
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Les solutions sont donc $\mathcal{S} = \left\{ \Var{x1|round(2)}; \Var{x2|round(2)} \right\}$
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\Block{elif Delta == 0}
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Alors $\Delta = \Var{Delta} = 0$ donc il y a une solution
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\Block{set x1 = Expression("-{b}/(2*{a})".format(b = R._coef[1], a = R._coef[2]))}
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\begin{eqnarray*}
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x_1 = \frac{-b}{2a} = \Var{" = ".join(x1.simplify())}
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\end{eqnarray*}
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Les solutions sont donc $\mathcal{S} = \left\{ \Var{x1.simplified()}\right\}$
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\Block{else}
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Alors $\Delta = \Var{Delta} < 0$ donc il n'y a pas de solution.
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\Block{endif}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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