2021-2022/2nd/01_Fraction_Developpement_Litteral/6E_bilan_dev_red.tex

\documentclass[a4paper,10pt]{article}
\usepackage{myXsim}
\usepackage{amsmath}

\author{Benjamin Bertrand}
\title{Information chiffrée 1 - Exercices}
\date{Octobre 2021}

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\begin{document}

\begin{exercise}[subtitle={Réductions}]
    Réduire les expressions suivantes
    \begin{multicols}{3}
        \begin{enumerate}[label={\Alph*=}]
            \item $7x + 1 + 7x + 8$
            \item $3x - 6 + 4x - 2$

            \item $- 4x^{2} - 6 - 6x^{2} + 6 + 6x + 8$
            \item $- 1x + 3 - 8x + 1 - 9x - 2x$

            \item $18x + 19 + 12x + 4x + 18$
            \item $- 3x - 7 + 3x + 9$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{3}
        \begin{flalign*}
            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 - 6 + 4x - 2\\ =& 3x - 6 + 4x - 2\\ =& 3x + 4x - 6 - 2\\ =& (3 + 4) \times x - 8\\ =& 7x - 8
        \end{flalign*}
        \begin{flalign*}
            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 =& - 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 =& 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 =& - 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 $- 3(- 5x - 6)$
            \item $10(6x + 4)$

            \item $5x(8x + 10)$
            \item $- 10x(- 6x - 9) - 9$
            \item $8x - 3x(- 6x - 7)$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{3}
        \begin{flalign*}
            A =& 10(x + 3)\\ =& 10x + 10 \times 3\\ =& 10x + 30
        \end{flalign*}
        \begin{flalign*}
            B =& - 3(- 5x - 6)\\ =& - 3 \times - 5x - 3(- 6)\\ =& - 3(- 5) \times x + 18\\ =& 15x + 18
        \end{flalign*}
        \begin{flalign*}
            C =& 10(6x + 4)\\ =& 10 \times 6x + 10 \times 4\\ =& 10 \times 6 \times x + 40\\ =& 60x + 40
        \end{flalign*}
        \begin{flalign*}
            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 =& - 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 =& 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 $(x + 4)(x - 4)$
            \item $(6x + 2)(8x - 9)$
            \item $(- 8x + 8)(- 2x + 2)$

            \item $(6x + 9)(10x + 10)$
            \item $(7x - 8)^{2}$
            \item $(10x - 7)^{2}$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{2}
        \begin{flalign*}
            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 =& (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 =& (- 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 =& (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 =& (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 =& (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}

\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 $\left(\dfrac{- 5}{- 4} x - 4\right)\left(3x + \dfrac{8}{10}\right)$

        \end{enumerate}
    \end{multicols}
\end{exercise}

\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}
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