2022-2023/2nd/03_Calcul_litteral/1_exercises.tex

\begin{exercise}[subtitle={Réduire - technique}, step={1}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
    Réduire les expressions suivantes

    \begin{multicols}{2}
        \begin{enumerate}
            \item $A = - 6x - 7 + 10x + 3$
            \item $B = - 4t - 3 - 10t - 7t$
            \item $C = - 8t - 4 - 3t + 8t$
            \item $D = 4x + 5 - 4x - 9$
            \item $E = - 7t + 9 + 2t - 9 - 4t$
            \item $F = \dfrac{- 9}{9} + 9a + 4a + 8$
            \item $G = 6x^{2} + 4 - 2x^{2} + 4 - 5x^{2}$
            \item $H = - 9x - 10 - 3x^{2} + 5 - 4x^{2}$
            \item $I = - 9x - 5 + 7x^{2} + 9x + 9x^{2}$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{3}
        \begin{enumerate}
            \item
                \begin{align*}
                    A & = - 6x - 7 + 10x + 3 \\ & = - 6x - 7 + 10x + 3 \\ & = - 6x + 10x - 7 + 3 \\ & = (- 6 + 10) \times x - 4 \\ & = 4x - 4
                \end{align*}
            \item
                \begin{align*}
                    B & = - 4t - 3 - 10t - 7t \\ & = - 4t - 3 + (- 10 - 7) \times t \\ & = - 4t - 3 - 17t \\ & = - 4t - 17t - 3 \\ & = (- 4 - 17) \times t - 3 \\ & = - 21t - 3
                \end{align*}
            \item
                \begin{align*}
                    C & = - 8t - 4 - 3t + 8t \\ & = - 8t - 4 + (- 3 + 8) \times t \\ & = - 8t - 4 + 5t \\ & = - 8t + 5t - 4 \\ & = (- 8 + 5) \times t - 4 \\ & = - 3t - 4
                \end{align*}
            \item
                \begin{align*}
                    D & = 4x + 5 - 4x - 9 \\ & = 4x + 5 - 4x - 9 \\ & = 4x - 4x + 5 - 9 \\ & = (4 - 4) \times x - 4 \\ & = 0x - 4 \\ & = - 4
                \end{align*}
            \item
                \begin{align*}
                    E & = - 7t + 9 + 2t - 9 - 4t \\ & = - 7t + 9 + (2 - 4) \times t - 9 \\ & = - 7t + 9 - 9 - 2t \\ & = (- 7 - 2) \times t + 0 \\ & = - 9t
                \end{align*}
            \item
                \begin{align*}
                    F & = \dfrac{- 9}{9} + 9a + 4a + 8 \\ & = 9a + \dfrac{- 9}{9} + 4a + 8 \\ & = 9a + 4a + \dfrac{- 9}{9} + 8 \\ & = (9 + 4) \times a + \dfrac{- 9}{9} + \dfrac{8}{1} \\ & = 13a + \dfrac{- 9}{9} + \dfrac{8 \times 9}{1 \times 9} \\ & = 13a + \dfrac{- 9}{9} + \dfrac{72}{9} \\ & = 13a + \dfrac{- 9}{9} + \dfrac{72}{9} \\ & = 13a + \dfrac{- 9 + 72}{9} \\ & = 13a + \dfrac{63}{9}
                \end{align*}
            \item
                \begin{align*}
                    G & = 6x^{2} + 4 - 2x^{2} + 4 - 5x^{2} \\ & = 6x^{2} + 4 + (- 2 - 5) \times x^{2} + 4 \\ & = 6x^{2} + 4 + 4 - 7x^{2} \\ & = (6 - 7) \times x^{2} + 8 \\ & = - x^{2} + 8
                \end{align*}
            \item
                \begin{align*}
                    H & = - 9x - 10 - 3x^{2} + 5 - 4x^{2} \\ & = - 3x^{2} - 9x - 10 + 5 - 4x^{2} \\ & = - 3x^{2} - 4x^{2} - 9x - 10 + 5 \\ & = (- 3 - 4) \times x^{2} - 9x - 5 \\ & = - 7x^{2} - 9x - 5
                \end{align*}
            \item
                \begin{align*}
                    I & = - 9x - 5 + 7x^{2} + 9x + 9x^{2} \\ & = 7x^{2} - 9x - 5 + 9x + 9x^{2} \\ & = 7x^{2} + 9x^{2} - 9x + 9x - 5 \\ & = (7 + 9) \times x^{2} + (- 9 + 9) \times x - 5 \\ & = 16x^{2} - 5
                \end{align*}
        \end{enumerate}
    \end{multicols}
\end{solution}

\begin{exercise}[subtitle={Développer 1 - technique}, step={2}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
    Développer puis réduire les expressions suivantes

    \begin{multicols}{2}
        \begin{enumerate}
            \item $A = - 6(3x - 7)$
            \item $B = - 6(- 7 + 3t)$
            \item $C = t(7 - 5t)$
            \item $D = 10x(4x + 7)$
            \item $E = - 3x(- 5x - 4)$
            \item $F = \dfrac{2}{10} \times x(2x + 9)$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{3}
        \begin{enumerate}
            \item
                \begin{align*}
                    A & = - 6(3x - 7) \\ & = - 6 \times 3x - 6(- 7) \\ & = - 6 \times 3 \times x + 42 \\ & = - 18x + 42
                \end{align*}
            \item
                \begin{align*}
                    B & = - 6(- 7 + 3t) \\ & = - 6 \times 3t - 6(- 7) \\ & = - 6 \times 3 \times t + 42 \\ & = - 18t + 42
                \end{align*}
            \item
                \begin{align*}
                    C & = t(7 - 5t) \\ & = t \times - 5t + t \times 7 \\ & = - 5t^{2} + 7t
                \end{align*}
            \item
                \begin{align*}
                    D & = 10x(4x + 7) \\ & = 10x \times 4x + 10x \times 7 \\ & = 10 \times 4 \times x^{1 + 1} + 7 \times 10 \times x \\ & = 40x^{2} + 70x
                \end{align*}
            \item
                \begin{align*}
                    E & = - 3x(- 5x - 4) \\ & = - 3x \times - 5x - 3x(- 4) \\ & = - 3(- 5) \times x^{1 + 1} - 4(- 3) \times x \\ & = 15x^{2} + 12x
                \end{align*}
            \item
                \begin{align*}
                    F & = \dfrac{2}{10} \times x(2x + 9) \\ & = \dfrac{2}{10} \times x \times 2x + \dfrac{2}{10} \times x \times 9 \\ & = \dfrac{2}{10} \times 2 \times x^{1 + 1} + 9 \times \dfrac{2}{10} \times x \\ & = \dfrac{2 \times 2}{10} \times x^{2} + \dfrac{9 \times 2}{10} \times x \\ & = \dfrac{4}{10} \times x^{2} + \dfrac{18}{10} \times x
                \end{align*}
        \end{enumerate}
    \end{multicols}
\end{solution}

\begin{exercise}[subtitle={Développer 2 - technique}, step={2}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
    Développer puis réduire les expressions suivantes

    \begin{multicols}{2}
        \begin{enumerate}
            \item $A = (- 10x - 9)(3x - 5)$
            \item $B = (8t - 6)(4t - 2)$
            \item $C = (2x + 6)(- 6x - 7)$
            \item $D = (2x - 9)(2x + 8)$
            \item $E = (- 3x + 2)^{2}$
            \item $F = (8x + 5)^{2}$
            \item $G = (8x + 5)^{2}$
            \item $H = (\dfrac{10}{7} \times x - 10)^{2}$
        \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{solution}
    \begin{multicols}{2}
        \begin{enumerate}
            \item
                \begin{align*}
                    A & = (- 10x - 9)(3x - 5) \\ & = - 10x \times 3x - 10x(- 5) - 9 \times 3x - 9(- 5) \\ & = - 10 \times 3 \times x^{1 + 1} - 5(- 10) \times x - 9 \times 3 \times x + 45 \\ & = 50x - 27x - 30x^{2} + 45 \\ & = (50 - 27) \times x - 30x^{2} + 45 \\ & = - 30x^{2} + 23x + 45
                \end{align*}
            \item
                \begin{align*}
                    B & = (8t - 6)(4t - 2) \\ & = 8t \times 4t + 8t(- 2) - 6 \times 4t - 6(- 2) \\ & = 8 \times 4 \times t^{1 + 1} - 2 \times 8 \times t - 6 \times 4 \times t + 12 \\ & = - 16t - 24t + 32t^{2} + 12 \\ & = (- 16 - 24) \times t + 32t^{2} + 12 \\ & = 32t^{2} - 40t + 12
                \end{align*}
            \item
                \begin{align*}
                    C & = (2x + 6)(- 6x - 7) \\ & = 2x \times - 6x + 2x(- 7) + 6 \times - 6x + 6(- 7) \\ & = 2(- 6) \times x^{1 + 1} - 7 \times 2 \times x + 6(- 6) \times x - 42 \\ & = - 14x - 36x - 12x^{2} - 42 \\ & = (- 14 - 36) \times x - 12x^{2} - 42 \\ & = - 12x^{2} - 50x - 42
                \end{align*}
            \item
                \begin{align*}
                    D & = (2x - 9)(2x + 8) \\ & = 2x \times 2x + 2x \times 8 - 9 \times 2x - 9 \times 8 \\ & = 2 \times 2 \times x^{1 + 1} + 8 \times 2 \times x - 9 \times 2 \times x - 72 \\ & = 16x - 18x + 4x^{2} - 72 \\ & = (16 - 18) \times x + 4x^{2} - 72 \\ & = 4x^{2} - 2x - 72
                \end{align*}
            \item
                \begin{align*}
                    E & = (- 3x + 2)^{2} \\ & = (- 3x + 2)(- 3x + 2) \\ & = - 3x \times - 3x - 3x \times 2 + 2 \times - 3x + 2 \times 2 \\ & = - 3(- 3) \times x^{1 + 1} + 2(- 3) \times x + 2(- 3) \times x + 4 \\ & = - 6x - 6x + 9x^{2} + 4 \\ & = (- 6 - 6) \times x + 9x^{2} + 4 \\ & = 9x^{2} - 12x + 4
                \end{align*}
            \item
                \begin{align*}
                    F & = (8x + 5)^{2} \\ & = (8x + 5)(8x + 5) \\ & = 8x \times 8x + 8x \times 5 + 5 \times 8x + 5 \times 5 \\ & = 8 \times 8 \times x^{1 + 1} + 5 \times 8 \times x + 5 \times 8 \times x + 25 \\ & = 40x + 40x + 64x^{2} + 25 \\ & = (40 + 40) \times x + 64x^{2} + 25 \\ & = 64x^{2} + 80x + 25
                \end{align*}
            \item
                \begin{align*}
                    G & = (8x + 5)^{2} \\ & = (8x + 5)(8x + 5) \\ & = 8x \times 8x + 8x \times 5 + 5 \times 8x + 5 \times 5 \\ & = 8 \times 8 \times x^{1 + 1} + 5 \times 8 \times x + 5 \times 8 \times x + 25 \\ & = 40x + 40x + 64x^{2} + 25 \\ & = (40 + 40) \times x + 64x^{2} + 25 \\ & = 64x^{2} + 80x + 25
                \end{align*}
            \item
                \begin{align*}
                    H & = (\dfrac{10}{7} \times x - 10)^{2} \\ & = (\dfrac{10}{7} \times x - 10)(\dfrac{10}{7} \times x - 10) \\ & = \dfrac{10}{7} \times x \times \dfrac{10}{7} \times x + \dfrac{10}{7} \times x(- 10) - 10 \times \dfrac{10}{7} \times x - 10(- 10) \\ & = \dfrac{10}{7} \times \dfrac{10}{7} \times x^{1 + 1} - 10 \times \dfrac{10}{7} \times x - 10 \times \dfrac{10}{7} \times x + 100 \\ & = \dfrac{- 10 \times 10}{7} \times x + \dfrac{- 10 \times 10}{7} \times x + \dfrac{10 \times 10}{7 \times 7} \times x^{2} + 100 \\ & = \dfrac{- 100}{7} \times x + \dfrac{100}{49} \times x^{2} + \dfrac{- 100}{7} \times x + 100 \\ & = 100 + \dfrac{100}{49} \times x^{2} + \dfrac{- 100}{7} \times x + \dfrac{- 100}{7} \times x \\ & = 100 + \dfrac{100}{49} \times x^{2} + (\dfrac{- 100}{7} + \dfrac{- 100}{7}) \times x \\ & = 100 + \dfrac{100}{49} \times x^{2} + \dfrac{- 100 - 100}{7} \times x \\ & = \dfrac{100}{49} \times x^{2} + \dfrac{- 200}{7} \times x + 100
                \end{align*}
        \end{enumerate}
    \end{multicols}
\end{solution}