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Feat: Ajout d'un exo avec des fractions
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Bertrand Benjamin 2021-10-22 07:16:27 +02:00
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2nd/01_Fraction_Developpement_Litteral/6E_bilan_dev_red.pdf
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2nd/01_Fraction_Developpement_Litteral/6E_bilan_dev_red.tex
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\title{Information chiffrée 1 - Exercices}
\date{Octobre 2021}
\pagestyle{empty}
\xsimsetup{
solution/print = false
}
\pagestyle{empty}
\begin{document}
\begin{exercise}[subtitle={Réductions}]
Réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
\item $- 3x - 10 + 4x + 7$
\item $3x + 2 - 9x - 7$
\item $7x + 1 + 7x + 8$
\item $3x - 6 + 4x - 2$
\item $- 9x^{2} - 7 - 7x^{2} + 6 + 6x - 7$
\item $- 7x - 1 - 7x + 8 + 4x + 3x$
\item $- 4x^{2} - 6 - 6x^{2} + 6 + 6x + 8$
\item $- 1x + 3 - 8x + 1 - 9x - 2x$
\item $3x + 15 + 9x + 18x + 17$
\item $- 4x - 5 + 4x + 1$
\item $18x + 19 + 12x + 4x + 18$
\item $- 3x - 7 + 3x + 9$
\end{enumerate}
\end{multicols}
\end{exercise}
@ -32,36 +33,37 @@
\begin{solution}
\begin{multicols}{3}
\begin{flalign*}
A =& - 3x - 10 + 4x + 7\\ =& - 3x - 10 + 4x + 7\\ =& - 3x + 4x - 10 + 7\\ =& (- 3 + 4) \times x - 3\\ =& x - 3
A =& 7x + 1 + 7x + 8\\ =& 7x + 1 + 7x + 8\\ =& 7x + 7x + 1 + 8\\ =& (7 + 7) \times x + 9\\ =& 14x + 9
\end{flalign*}
\begin{flalign*}
B =& 3x + 2 - 9x - 7\\ =& 3x + 2 - 9x - 7\\ =& 3x - 9x + 2 - 7\\ =& (3 - 9) \times x - 5\\ =& - 6x - 5
B =& 3x - 6 + 4x - 2\\ =& 3x - 6 + 4x - 2\\ =& 3x + 4x - 6 - 2\\ =& (3 + 4) \times x - 8\\ =& 7x - 8
\end{flalign*}
\begin{flalign*}
C =& - 9x^{2} - 7 - 7x^{2} + 6 + 6x - 7\\ =& - 9x^{2} - 7x^{2} - 7 - 1 + 6x\\ =& (- 9 - 7) \times x^{2} + 6x - 7 - 1\\ =& - 16x^{2} + 6x - 8
C =& - 4x^{2} - 6 - 6x^{2} + 6 + 6x + 8\\ =& - 4x^{2} - 6x^{2} - 6 + 14 + 6x\\ =& (- 4 - 6) \times x^{2} + 6x - 6 + 14\\ =& - 10x^{2} + 6x + 8
\end{flalign*}
\begin{flalign*}
D =& - 7x - 1 - 7x + 8 + 4x + 3x\\ =& - 7x - 1 + 8 - 7x + (4 + 3) \times x\\ =& (- 7 - 7) \times x + 7 + 7x\\ =& - 14x + 7 + 7x\\ =& - 14x + 7x + 7\\ =& (- 14 + 7) \times x + 7\\ =& - 7x + 7
D =& - 1x + 3 - 8x + 1 - 9x - 2x\\ =& - x + 3 + 1 - 8x + (- 9 - 2) \times x\\ =& (- 1 - 8) \times x + 4 - 11x\\ =& - 9x + 4 - 11x\\ =& - 9x - 11x + 4\\ =& (- 9 - 11) \times x + 4\\ =& - 20x + 4
\end{flalign*}
\begin{flalign*}
E =& 3x + 15 + 9x + 18x + 17\\ =& 3x + 15 + (9 + 18) \times x + 17\\ =& 3x + 15 + 17 + 27x\\ =& (3 + 27) \times x + 32\\ =& 30x + 32
E =& 18x + 19 + 12x + 4x + 18\\ =& 18x + 19 + (12 + 4) \times x + 18\\ =& 18x + 19 + 18 + 16x\\ =& (18 + 16) \times x + 37\\ =& 34x + 37
\end{flalign*}
\begin{flalign*}
F =& - 4x - 5 + 4x + 1\\ =& - 4x - 5 + 4x + 1\\ =& - 4x + 4x - 5 + 1\\ =& (- 4 + 4) \times x - 4\\ =& 0x - 4\\ =& - 4
F =& - 3x - 7 + 3x + 9\\ =& - 3x - 7 + 3x + 9\\ =& - 3x + 3x - 7 + 9\\ =& (- 3 + 3) \times x + 2\\ =& 0x + 2\\ =& 2
\end{flalign*}
\end{multicols}
\end{solution}
\begin{exercise}[subtitle={Simple développement}]
Développer puis réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
\item $10(x + 3)$
\item $8(- 10x + 2)$
\item $3(- 9x + 3)$
\item $- 3(- 5x - 6)$
\item $10(6x + 4)$
\item $- 10x(- 6x + 3)$
\item $2x(10x - 2) - 4$
\item $- 3x - 7x(- 10x + 9)$
\item $5x(8x + 10)$
\item $- 10x(- 6x - 9) - 9$
\item $8x - 3x(- 6x - 7)$
\end{enumerate}
\end{multicols}
\end{exercise}
@ -72,70 +74,89 @@
A =& 10(x + 3)\\ =& 10x + 10 \times 3\\ =& 10x + 30
\end{flalign*}
\begin{flalign*}
B =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
B =& - 3(- 5x - 6)\\ =& - 3 \times - 5x - 3(- 6)\\ =& - 3(- 5) \times x + 18\\ =& 15x + 18
\end{flalign*}
\begin{flalign*}
C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
C =& 10(6x + 4)\\ =& 10 \times 6x + 10 \times 4\\ =& 10 \times 6 \times x + 40\\ =& 60x + 40
\end{flalign*}
\begin{flalign*}
D =& - 10x(- 6x + 3)\\ =& - 10x \times - 6x - 10x \times 3\\ =& - 10(- 6) \times x^{1 + 1} + 3(- 10) \times x\\ =& 60x^{2} - 30x
D =& 5x(8x + 10)\\ =& 5x \times 8x + 5x \times 10\\ =& 5 \times 8 \times x^{1 + 1} + 10 \times 5 \times x\\ =& 40x^{2} + 50x
\end{flalign*}
\begin{flalign*}
E =& 2x(10x - 2) - 4\\ =& 2x \times 10x + 2x(- 2) - 4\\ =& 2 \times 10 \times x^{1 + 1} - 2 \times 2 \times x - 4\\ =& 20x^{2} - 4x - 4
E =& - 10x(- 6x - 9) - 9\\ =& - 10x \times - 6x - 10x(- 9) - 9\\ =& - 10(- 6) \times x^{1 + 1} - 9(- 10) \times x - 9\\ =& 60x^{2} + 90x - 9
\end{flalign*}
\begin{flalign*}
F =& - 3x - 7x(- 10x + 9)\\ =& - 3x - 7x \times - 10x - 7x \times 9\\ =& - 3x - 7(- 10) \times x^{1 + 1} + 9(- 7) \times x\\ =& - 3x - 63x + 70x^{2}\\ =& (- 3 - 63) \times x + 70x^{2}\\ =& 70x^{2} - 66x
F =& 8x - 3x(- 6x - 7)\\ =& 8x - 3x \times - 6x - 3x(- 7)\\ =& 8x - 3(- 6) \times x^{1 + 1} - 7(- 3) \times x\\ =& 8x + 21x + 18x^{2}\\ =& (8 + 21) \times x + 18x^{2}\\ =& 18x^{2} + 29x
\end{flalign*}
\end{multicols}
\end{solution}
\begin{exercise}[subtitle={Double développement}]
Développer puis réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
\item $(x + 10)(x + 6)$
\item $(8x - 6)(- 5x - 10)$
\item $(- 6x + 9)(7x - 3)$
\item $(x + 4)(x - 4)$
\item $(6x + 2)(8x - 9)$
\item $(- 8x + 8)(- 2x + 2)$
\item $(5x + 2)(- 6x + 2)$
\item $(- 8x + 8)^{2}$
\item $(2x - 6)^{2}$
\item $(6x + 9)(10x + 10)$
\item $(7x - 8)^{2}$
\item $(10x - 7)^{2}$
\end{enumerate}
\end{multicols}
\end{exercise}
\begin{solution}
\begin{multicols}{3}
\begin{multicols}{2}
\begin{flalign*}
A =& 10(x + 3)\\ =& 10x + 10 \times 3\\ =& 10x + 30
A =& (x + 4)(x - 4)\\ =& x \times x + x(- 4) + 4x + 4(- 4)\\ =& x^{2} - 16 + (- 4 + 4) \times x\\ =& x^{2} - 16
\end{flalign*}
\begin{flalign*}
B =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
B =& (6x + 2)(8x - 9)\\ =& 6x \times 8x + 6x(- 9) + 2 \times 8x + 2(- 9)\\ =& 6 \times 8 \times x^{1 + 1} - 9 \times 6 \times x + 2 \times 8 \times x - 18\\ =& - 54x + 16x + 48x^{2} - 18\\ =& (- 54 + 16) \times x + 48x^{2} - 18\\ =& 48x^{2} - 38x - 18
\end{flalign*}
\begin{flalign*}
C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
C =& (- 8x + 8)(- 2x + 2)\\ =& - 8x \times - 2x - 8x \times 2 + 8 \times - 2x + 8 \times 2\\ =& - 8(- 2) \times x^{1 + 1} + 2(- 8) \times x + 8(- 2) \times x + 16\\ =& - 16x - 16x + 16x^{2} + 16\\ =& (- 16 - 16) \times x + 16x^{2} + 16\\ =& 16x^{2} - 32x + 16
\end{flalign*}
\begin{flalign*}
D =& - 10x(- 6x + 3)\\ =& - 10x \times - 6x - 10x \times 3\\ =& - 10(- 6) \times x^{1 + 1} + 3(- 10) \times x\\ =& 60x^{2} - 30x
D =& (6x + 9)(10x + 10)\\ =& 6x \times 10x + 6x \times 10 + 9 \times 10x + 9 \times 10\\ =& 6 \times 10 \times x^{1 + 1} + 10 \times 6 \times x + 9 \times 10 \times x + 90\\ =& 60x + 90x + 60x^{2} + 90\\ =& (60 + 90) \times x + 60x^{2} + 90\\ =& 60x^{2} + 150x + 90
\end{flalign*}
\begin{flalign*}
E =& 2x(10x - 2) - 4\\ =& 2x \times 10x + 2x(- 2) - 4\\ =& 2 \times 10 \times x^{1 + 1} - 2 \times 2 \times x - 4\\ =& 20x^{2} - 4x - 4
E =& (7x - 8)^{2}\\ =& (7x - 8)(7x - 8)\\ =& 7x \times 7x + 7x(- 8) - 8 \times 7x - 8(- 8)\\ =& 7 \times 7 \times x^{1 + 1} - 8 \times 7 \times x - 8 \times 7 \times x + 64\\ =& - 56x - 56x + 49x^{2} + 64\\ =& (- 56 - 56) \times x + 49x^{2} + 64\\ =& 49x^{2} - 112x + 64
\end{flalign*}
\begin{flalign*}
F =& - 3x - 7x(- 10x + 9)\\ =& - 3x - 7x \times - 10x - 7x \times 9\\ =& - 3x - 7(- 10) \times x^{1 + 1} + 9(- 7) \times x\\ =& - 3x - 63x + 70x^{2}\\ =& (- 3 - 63) \times x + 70x^{2}\\ =& 70x^{2} - 66x
F =& (10x - 7)^{2}\\ =& (10x - 7)(10x - 7)\\ =& 10x \times 10x + 10x(- 7) - 7 \times 10x - 7(- 7)\\ =& 10 \times 10 \times x^{1 + 1} - 7 \times 10 \times x - 7 \times 10 \times x + 49\\ =& - 70x - 70x + 100x^{2} + 49\\ =& (- 70 - 70) \times x + 100x^{2} + 49\\ =& 100x^{2} - 140x + 49
\end{flalign*}
\end{multicols}
\end{solution}
\vfill
\begin{exercise}[subtitle={Double développement}]
Développer puis réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
\item $2x + \dfrac{- 8}{7} - 8x + \dfrac{- 5}{7}$
\item $8\left(- 4x + \dfrac{3}{5}\right)$
\item $\left(\dfrac{- 5}{- 4} x - 4\right)\left(3x + \dfrac{8}{10}\right)$
\printexercise{exercise}{1}
\printexercise{exercise}{2}
\printexercise{exercise}{3}
\vfill
\end{enumerate}
\end{multicols}
\end{exercise}
\printexercise{exercise}{1}
\printexercise{exercise}{2}
\printexercise{exercise}{3}
\begin{solution}
\begin{multicols}{2}
\begin{flalign*}
A =& 2x + \dfrac{- 8}{7} - 8x + \dfrac{- 5}{7}\\ =& 2x + \dfrac{- 8}{7} + \dfrac{- 5}{7} - 8x\\ =& (2 - 8) \times x + \dfrac{- 8 - 5}{7}\\ =& - 6x + \dfrac{- 13}{7}
\end{flalign*}
\begin{flalign*}
B =& 8(- 4x + \dfrac{3}{5})\\ =& 8 \times - 4x + 8 \times \dfrac{3}{5}\\ =& 8(- 4) \times x + \dfrac{8 \times 3}{5}\\ =& - 32x + \dfrac{24}{5}
\end{flalign*}
\begin{flalign*}
C =& \left(\dfrac{- 5}{- 4} \times x - 4\right)\left(3x + \dfrac{8}{10}\right)\\ =& \dfrac{- 5}{- 4} \times x \times 3x + \dfrac{- 5}{- 4} \times x \times \dfrac{8}{10} - 4 \times 3x - 4 \times \dfrac{8}{10}\\ =& \dfrac{- 5}{- 4} \times 3 \times x^{1 + 1} + \dfrac{8}{10} \times \dfrac{- 5}{- 4} \times x - 4 \times 3 \times x + \dfrac{- 4 \times 8}{10}\\ =& \dfrac{- 5 \times 3}{- 4} \times x^{2} + \dfrac{8\left(- 5\right)}{10\left(- 4\right)} \times x - 12x + \dfrac{- 32}{10}\\ =& \dfrac{- 40}{- 40} \times x + \dfrac{- 15}{- 4} \times x^{2} - 12x + \dfrac{- 32}{10}\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{- 40}{- 40} \times x - 12x + \dfrac{- 32}{10}\\ =& \dfrac{- 15}{- 4} \times x^{2} + \left(\dfrac{- 40}{- 40} - 12\right) \times x + \dfrac{- 32}{10}\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{- 32}{10} + \left(\dfrac{- 40}{- 40} + \dfrac{- 12}{1}\right) \times x\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{- 32}{10} + \left(\dfrac{- 40}{- 40} + \dfrac{- 12\left(- 40\right)}{1\left(- 40\right)}\right) \times x\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{- 32}{10} + \left(\dfrac{- 40}{- 40} + \dfrac{480}{- 40}\right) \times x\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{- 32}{10} + \dfrac{- 40 + 480}{- 40} \times x\\ =& \dfrac{- 15}{- 4} \times x^{2} + \dfrac{440}{- 40} \times x + \dfrac{- 32}{10}
\end{flalign*}
\end{multicols}
\end{solution}
\vfill
\printexercise{exercise}{1,2,3,4}
\vfill
\newpage

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2nd/01_Fraction_Developpement_Litteral/tpl_6E_bilan_dev_red.tex
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@ -122,6 +122,58 @@
\end{multicols}
\end{exercise}
\begin{solution}
\begin{multicols}{3}
\begin{flalign*}
A =& \Var{a.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
B =& \Var{b.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
C =& \Var{c.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
D =& \Var{d.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
E =& \Var{e.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
F =& \Var{f.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\end{multicols}
\end{solution}
\begin{exercise}[subtitle={Double développement}]
Développer puis réduire les expressions suivantes
\begin{multicols}{3}
\begin{enumerate}[label={\Alph*=}]
%- set a = Expression.random("{a}*x + {c}/{b} + {c}*x + {d}/{b}", rejected=[-1,0,1])
\item $\Var{a}$
%- set b = Expression.random("{c}({a}x + {b}/{d})", rejected=[-1,0,1])
\item $\Var{b}$
%- set c = Expression.random("({a}/{b}x + {b})*({c}x + {d}/{e})", rejected=[-1,0,1])
\item $\Var{c}$
\end{enumerate}
\end{multicols}
\end{exercise}
\begin{solution}
\begin{multicols}{3}
\begin{flalign*}
A =& \Var{a.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
B =& \Var{b.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\begin{flalign*}
C =& \Var{c.simplify().explain() | join('\\\ =& ')}
\end{flalign*}
\end{multicols}
\end{solution}
\printcollection{exercise}
\newpage