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

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

\pagestyle{empty}

\xsimsetup{
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}

\begin{document}

\begin{exercise}[subtitle={Réductions}]
    \begin{multicols}{3}
        \begin{enumerate}[label={\Alph*=}]
            \item $- 3x - 10 + 4x + 7$
            \item $3x + 2 - 9x - 7$

            \item $- 9x^{2} - 7 - 7x^{2} + 6 + 6x - 7$
            \item $- 7x - 1 - 7x + 8 + 4x + 3x$

            \item $3x + 15 + 9x + 18x + 17$
            \item $- 4x - 5 + 4x + 1$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\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
        \end{flalign*}
        \begin{flalign*}
            B =& 3x + 2 - 9x - 7\\ =& 3x + 2 - 9x - 7\\ =& 3x - 9x + 2 - 7\\ =& (3 - 9) \times x - 5\\ =& - 6x - 5
        \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
        \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
        \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
        \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
        \end{flalign*}
    \end{multicols}
\end{solution}

\begin{exercise}[subtitle={Simple développement}]
    \begin{multicols}{3}
        \begin{enumerate}[label={\Alph*=}]
            \item $10(x + 3)$
            \item $8(- 10x + 2)$
            \item $3(- 9x + 3)$

            \item $- 10x(- 6x + 3)$
            \item $2x(10x - 2) - 4$
            \item $- 3x - 7x(- 10x + 9)$
        \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 =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
        \end{flalign*}
        \begin{flalign*}
            C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
        \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
        \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
        \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
        \end{flalign*}
    \end{multicols}
\end{solution}

\begin{exercise}[subtitle={Double développement}]
    \begin{multicols}{3}
        \begin{enumerate}[label={\Alph*=}]
            \item $(x + 10)(x + 6)$
            \item $(8x - 6)(- 5x - 10)$
            \item $(- 6x + 9)(7x - 3)$

            \item $(5x + 2)(- 6x + 2)$
            \item $(- 8x + 8)^{2}$
            \item $(2x - 6)^{2}$
        \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 =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
        \end{flalign*}
        \begin{flalign*}
            C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
        \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
        \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
        \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
        \end{flalign*}
    \end{multicols}
\end{solution}

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\end{document}
