From 8b9a95cea8d193b962d762a57a36a67e6e9c65c9 Mon Sep 17 00:00:00 2001
From: Bertrand Benjamin <benjamin.bertrand@opytex.org>
Date: Thu, 12 Jan 2023 10:26:01 +0100
Subject: [PATCH] Feat(2nd): premier jet sur la factorisation

---
 .../1E_factorisation.pdf                      | Bin 0 -> 30550 bytes
 .../1E_factorisation.tex                      |  18 ++
 .../exercises.tex                             | 185 ++++++++++++++++++
 .../index.rst                                 |  28 +++
 4 files changed, 231 insertions(+)
 create mode 100644 2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.pdf
 create mode 100644 2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.tex
 create mode 100644 2nd/10_Factorisation_et_identites_remarquables/exercises.tex
 create mode 100644 2nd/10_Factorisation_et_identites_remarquables/index.rst

diff --git a/2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.pdf b/2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.pdf
new file mode 100644
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diff --git a/2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.tex b/2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.tex
new file mode 100644
index 0000000..7e42e28
--- /dev/null
+++ b/2nd/10_Factorisation_et_identites_remarquables/1E_factorisation.tex
@@ -0,0 +1,18 @@
+\documentclass[a4paper,10pt]{article}
+\usepackage{myXsim}
+
+\author{Benjamin Bertrand}
+\title{Identites remarquables racine carre - Exercices}
+\date{mai 2022}
+
+\DeclareExerciseCollection[step=1]{banque}
+\xsimsetup{collect}
+
+\pagestyle{empty}
+
+\begin{document}
+\input{exercises.tex}
+
+\printcollection{banque}
+
+\end{document}
diff --git a/2nd/10_Factorisation_et_identites_remarquables/exercises.tex b/2nd/10_Factorisation_et_identites_remarquables/exercises.tex
new file mode 100644
index 0000000..0de539f
--- /dev/null
+++ b/2nd/10_Factorisation_et_identites_remarquables/exercises.tex
@@ -0,0 +1,185 @@
+\begin{exercise}[subtitle={Factorisation simple}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    \begin{multicols}{2}
+        \begin{enumerate}
+            \item Relier les expressions égales entre elles puis écrire les égalités obtenues.
+
+                \begin{minipage}[c]{0.4\linewidth}
+                    \flushright
+                    $4x^2 + 4x \qquad \bullet$ \\[0.5cm]
+                    $48x + 9x^2 \qquad  \bullet$ \\[0.5cm]
+                    $6x^2 - 4x \qquad \bullet$ \\[0.5cm]
+
+                \end{minipage}
+                \hfill
+                \begin{minipage}[c]{0.4\linewidth}
+                    \begin{itemize}
+                        \item $4x(x + 1)$
+                        \item $-2x(-3x + 2)$
+                        \item $4x(x + 4)$
+                        \item $9x(48x + 1)$
+                        \item $x(48x + 9)$
+                        \item $2x(3x - 2)$
+                    \end{itemize}
+                \end{minipage}
+            \item Factoriser les expressions suivantes
+                \begin{multicols}{2}
+                    \begin{enumerate}[label={\Alph* = }]
+                        \item $3x^2 + 4x$
+                        \item $8x + 4x^2$
+                        \item $x^2 + x$
+                        \item $4x^2 - 12x$
+
+                        \item $(x+2)^2 - 4$
+                        \item $2(x+1) + x(x+1)$
+                        \item $(2x-1)x - (2x+1)4$
+                    \end{enumerate}
+                \end{multicols}
+                \begin{enumerate}[label={\Alph* = }]
+                    \setcounter{enumii}{7}
+                    \item $(10x-1)(x+1) + (10x+1)(3x-1)$
+                    \item $(7x+1)(2x-1) - (7x+1)(3x-1)$
+                \end{enumerate}
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
+
+\begin{exercise}[subtitle={Identités remarquables}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    \begin{multicols}{2}
+        \begin{enumerate}
+            \item Relier les expressions égales entre elles puis écrire les égalités obtenues.
+
+                \begin{minipage}[c]{0.4\linewidth}
+                    \flushright
+                    $4x^2 + 4x + 1 \qquad \bullet$ \\[0.5cm]
+                    $64x^2 - 48x + 9 \qquad  \bullet$ \\[0.5cm]
+                    $36x^2 + 60x + 25 \qquad \bullet$ \\[0.5cm]
+                    $36x^2 - 60x + 25 \qquad \bullet$
+
+                \end{minipage}
+                \hfill
+                \begin{minipage}[c]{0.4\linewidth}
+                    \begin{itemize}
+                        \item $(8x - 3)^2$
+                        \item $(6x + 5)^2$
+                        \item $(2x + 1)^2$
+                        \item $(6x - 5)^2$
+                        \item $(36x + 25)^2$
+                        \item $(4x + 1)^2$
+                        \item $(2x - 1)^2$
+                        \item $(8x + 3)^2$
+                    \end{itemize}
+
+                \end{minipage}
+            \item (\groupMode) Chercher le lien entre les nombres des deux parties des égalités. Proposer une méthode pour Factoriser ce type d'expression.
+            \item (\groupMode) Factoriser les expressions suivantes
+                \begin{multicols}{2}
+                    \begin{enumerate}[label={\Alph* = }]
+                        \item $25x^2 + 20x + 4$
+                        \item $16x^2 + 40x + 25$
+                        \item $25x^2 - 20x + 4$
+                        \item $49x^2 + 112x + 64$
+                    \end{enumerate}
+                \end{multicols}
+
+                    \columnbreak
+
+            \item Relier les expressions égales entre elles puis écrire les égalités obtenues.
+
+                \begin{minipage}[c]{0.4\linewidth}
+                    \flushright
+                    $4x^2 - 9 \qquad \bullet$ \\[0.5cm]
+                    $64x^2 - 16 \qquad  \bullet$ \\[0.5cm]
+                    $49x^2  - 81\qquad \bullet$ \\[0.5cm]
+                    $36 - 9x^2 \qquad \bullet$
+                \end{minipage}
+                \hfill
+                \begin{minipage}[c]{0.4\linewidth}
+                    \begin{itemize}
+                        \item $(4x - 9)^2$
+                        \item $(3x + 6)(3x - 6)$
+                        \item $(7x + 9)(9 - 7x)$
+                        \item $(8x + 4)^2$
+                        \item $(2x + 3)(2x - 3)$
+                        \item $(4x + 9)(4x - 9)$
+                        \item $(7x + 9)(7x - 9)$
+                        \item $(8x - 4)(8x + 4)$
+                        \item $(6 - 3x)(6 + 3x)$
+                    \end{itemize}
+                \end{minipage}
+
+            \item (\groupMode) Chercher le lien entre les nombres des deux parties des égalités. Proposer une méthode pour Factoriser ce type d'expression.
+            \item (\groupMode) Factoriser les expressions suivantes
+                \begin{multicols}{2}
+                    \begin{enumerate}[label={\Alph* = }]
+                        \item $4x^2-9$
+                        \item $9x^2-25$
+                        \item $64x^2 - 1$
+                        \item $x^2 - 16$
+                    \end{enumerate}
+                \end{multicols}
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
+
+\begin{exercise}[subtitle={Factorisation}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    Factoriser les expressions suivantes quand c'est possible
+    \begin{multicols}{4}
+        \begin{enumerate}[label={\Alph* = }]
+            \item $4x^2 + 2x + 1$
+            \item $16x^2 - 1$
+
+            \item $x^2 - 4x + 4$
+            \item $x^2 + 10x + 25$
+
+            \item $121x - 22x + 1$
+            \item $81 + x^2$
+
+            \item $4x^2 + 49$
+            \item $64 - x^2$
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
+
+\begin{exercise}[subtitle={Équations}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    Résoudre les équations suivantes
+    \begin{multicols}{4}
+        \begin{enumerate}[label={\alph*) }]
+            \item $(2x+1)(x-2) = 0$
+            \item $(4x-2)^2 = 0$
+
+            \item $16x^2 - 1 = 0$
+            \item $4x^2 + 2x + 1 = 0$
+
+            \item $9x^2 - 6x + 1 = 0$
+            \item $x^2 - 16 = 0$
+
+            \item $x^2 - \dfrac{1}{4} = 0$
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
+
+
+\begin{exercise}[subtitle={Inéquations}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    Résoudre les inéquations suivantes
+    \begin{multicols}{4}
+        \begin{enumerate}[label={\alph*) }]
+            \item $(2x+1)(x-3) > 0$
+            \item $10x^2 - 1 < 0$
+            \item $4x^2 - 12x + 9 \leq 0$
+            \item $121 - x^2 > 0$
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
+
+
+\begin{exercise}[subtitle={Simplification de fraction}, step={1}, origin={Création}, topics={ Identites remarquables racine carre et puissance }, tags={ Calcul littéral, Racine carré, Puissance }]
+    Simplifier les fractions suivantes
+    \begin{multicols}{4}
+        \begin{enumerate}[label={\Alph* = }]
+            \item $\dfrac{x^2 + x}{x}$
+            \item $\dfrac{x^2 - 1}{x-1}$
+            \item $\dfrac{4x^2 - 28x + 49}{2x + 7}$
+            \item $\dfrac{36 - x^2}{x-6}$
+        \end{enumerate}
+    \end{multicols}
+\end{exercise}
diff --git a/2nd/10_Factorisation_et_identites_remarquables/index.rst b/2nd/10_Factorisation_et_identites_remarquables/index.rst
new file mode 100644
index 0000000..55d3597
--- /dev/null
+++ b/2nd/10_Factorisation_et_identites_remarquables/index.rst
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+Factorisation et identités remarquables
+#######################################
+
+:date: 2023-01-12
+:modified: 2023-01-12
+:authors: Benjamin Bertrand
+:tags: Calcul littéral
+:category: 2nd
+:summary: Factoriser des expressions avec et sans identités remarquables.
+
+
+Éléments du programme
+=====================
+
+Contenus
+--------
+
+Capacités attendues
+-------------------
+
+Commentaires
+------------
+
+Progression
+===========
+
+Étape 1:
+--------