163 lines
9.7 KiB
TeX
163 lines
9.7 KiB
TeX
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\begin{exercise}[subtitle={Réduire - technique}, step={1}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
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Réduire les expressions suivantes
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\begin{multicols}{2}
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\begin{enumerate}
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\item $A = 3x - 7 + 10x - 6$
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\item $B = - 7t - 3 - 10t - 4t$
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\item $C = 8t - 4 - 3t - 8t$
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\item $D = - 9x + 2 + 9x - 4$
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\item $E = 6t - 4 + 4t + 4 + 6t$
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\item $F = \dfrac{- 3}{3} + 4a - 7a - 2$
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\item $G = 8x^{2} + 10 + 9x^{2} - 3 - 6x^{2}$
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\item $H = - 8x + 10 - 4x^{2} - 5 + 4x^{2}$
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\item $I = 5x - 3 + 3x^{2} - 5x - 7x^{2}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{3}
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\begin{enumerate}
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\item
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\begin{align*}
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A & = 3x - 7 + 10x - 6 \\ & = 3x - 7 + 10x - 6 \\ & = 3x + 10x - 7 - 6 \\ & = (3 + 10) \times x - 13 \\ & = 13x - 13
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\end{align*}
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\item
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\begin{align*}
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B & = - 7t - 3 - 10t - 4t \\ & = - 7t - 3 + (- 10 - 4) \times t \\ & = - 7t - 3 - 14t \\ & = - 7t - 14t - 3 \\ & = (- 7 - 14) \times t - 3 \\ & = - 21t - 3
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\end{align*}
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\item
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\begin{align*}
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C & = 8t - 4 - 3t - 8t \\ & = 8t - 4 + (- 3 - 8) \times t \\ & = 8t - 4 - 11t \\ & = 8t - 11t - 4 \\ & = (8 - 11) \times t - 4 \\ & = - 3t - 4
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\end{align*}
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\item
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\begin{align*}
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D & = - 9x + 2 + 9x - 4 \\ & = - 9x + 2 + 9x - 4 \\ & = - 9x + 9x + 2 - 4 \\ & = (- 9 + 9) \times x - 2 \\ & = 0x - 2 \\ & = - 2
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\end{align*}
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\item
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\begin{align*}
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E & = 6t - 4 + 4t + 4 + 6t \\ & = 6t - 4 + (4 + 6) \times t + 4 \\ & = 6t - 4 + 4 + 10t \\ & = (6 + 10) \times t + 0 \\ & = 16t
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\end{align*}
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\item
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\begin{align*}
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F & = \dfrac{- 3}{3} + 4a - 7a - 2 \\ & = 4a + \dfrac{- 3}{3} - 7a - 2 \\ & = 4a - 7a + \dfrac{- 3}{3} - 2 \\ & = (4 - 7) \times a + \dfrac{- 3}{3} + \dfrac{- 2}{1} \\ & = - 3a + \dfrac{- 3}{3} + \dfrac{- 2 \times 3}{1 \times 3} \\ & = - 3a + \dfrac{- 3}{3} + \dfrac{- 6}{3} \\ & = - 3a + \dfrac{- 3}{3} + \dfrac{- 6}{3} \\ & = - 3a + \dfrac{- 3 - 6}{3} \\ & = - 3a + \dfrac{- 9}{3}
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\end{align*}
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\item
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\begin{align*}
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G & = 8x^{2} + 10 + 9x^{2} - 3 - 6x^{2} \\ & = 8x^{2} + 10 + (9 - 6) \times x^{2} - 3 \\ & = 8x^{2} + 10 - 3 + 3x^{2} \\ & = (8 + 3) \times x^{2} + 7 \\ & = 11x^{2} + 7
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\end{align*}
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\item
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\begin{align*}
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H & = - 8x + 10 - 4x^{2} - 5 + 4x^{2} \\ & = - 4x^{2} - 8x + 10 - 5 + 4x^{2} \\ & = - 4x^{2} + 4x^{2} - 8x + 10 - 5 \\ & = (- 4 + 4) \times x^{2} - 8x + 5 \\ & = - 8x + 5
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\end{align*}
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\item
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\begin{align*}
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I & = 5x - 3 + 3x^{2} - 5x - 7x^{2} \\ & = 3x^{2} + 5x - 3 - 5x - 7x^{2} \\ & = 3x^{2} - 7x^{2} + 5x - 5x - 3 \\ & = (3 - 7) \times x^{2} + (5 - 5) \times x - 3 \\ & = - 4x^{2} - 3
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\end{align*}
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\end{enumerate}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Développer 1 - technique}, step={2}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
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Réduire les expressions suivantes
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\begin{multicols}{2}
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\begin{enumerate}
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\item $A = 10(- 8x + 8)$
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\item $B = 7(- 4 + 8t)$
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\item $C = t(3 + 7t)$
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\item $D = - 9x(7x - 3)$
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\item $E = 5x(10x - 5)$
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\item $F = \dfrac{9}{4} \times x(2x + 8)$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{3}
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\begin{enumerate}
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\item
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\begin{align*}
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A & = 10(- 8x + 8) \\ & = 10 \times - 8x + 10 \times 8 \\ & = 10(- 8) \times x + 80 \\ & = - 80x + 80
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\end{align*}
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\item
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\begin{align*}
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B & = 7(- 4 + 8t) \\ & = 7 \times 8t + 7(- 4) \\ & = 7 \times 8 \times t - 28 \\ & = 56t - 28
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\end{align*}
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\item
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\begin{align*}
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C & = t(3 + 7t) \\ & = t \times 7t + t \times 3 \\ & = 7t^{2} + 3t
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\end{align*}
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\item
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\begin{align*}
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D & = - 9x(7x - 3) \\ & = - 9x \times 7x - 9x(- 3) \\ & = - 9 \times 7 \times x^{1 + 1} - 3(- 9) \times x \\ & = - 63x^{2} + 27x
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\end{align*}
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\item
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\begin{align*}
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E & = 5x(10x - 5) \\ & = 5x \times 10x + 5x(- 5) \\ & = 5 \times 10 \times x^{1 + 1} - 5 \times 5 \times x \\ & = 50x^{2} - 25x
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\end{align*}
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\item
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\begin{align*}
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F & = \dfrac{9}{4} \times x(2x + 8) \\ & = \dfrac{9}{4} \times x \times 2x + \dfrac{9}{4} \times x \times 8 \\ & = \dfrac{9}{4} \times 2 \times x^{1 + 1} + 8 \times \dfrac{9}{4} \times x \\ & = \dfrac{9 \times 2}{4} \times x^{2} + \dfrac{8 \times 9}{4} \times x \\ & = \dfrac{18}{4} \times x^{2} + \dfrac{72}{4} \times x
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\end{align*}
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\end{enumerate}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Développer 2 - technique}, step={2}, origin={D'anciennes choses}, topics={ Fraction Developpement Litteral }, tags={ Fractions, Developpement }, mode={\trainMode}]
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Réduire les expressions suivantes
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\begin{multicols}{2}
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\begin{enumerate}
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\item $A = (- 8x - 3)(- 7x - 10)$
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\item $B = (- 2t + 10)(2t + 6)$
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\item $C = (- 9x - 5)(3x + 4)$
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\item $D = (6x - 2)(9x + 10)$
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\item $E = (- 8x - 4)^{2}$
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\item $F = (- 8x - 10)^{2}$
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\item $G = (- 10x + 6)^{2}$
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\item $H = (\dfrac{- 6}{4} \times x - 7)^{2}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{enumerate}
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\item
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\begin{align*}
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A & = (- 8x - 3)(- 7x - 10) \\ & = - 8x \times - 7x - 8x(- 10) - 3 \times - 7x - 3(- 10) \\ & = - 8(- 7) \times x^{1 + 1} - 10(- 8) \times x - 3(- 7) \times x + 30 \\ & = 80x + 21x + 56x^{2} + 30 \\ & = (80 + 21) \times x + 56x^{2} + 30 \\ & = 56x^{2} + 101x + 30
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\end{align*}
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\item
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\begin{align*}
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B & = (- 2t + 10)(2t + 6) \\ & = - 2t \times 2t - 2t \times 6 + 10 \times 2t + 10 \times 6 \\ & = - 2 \times 2 \times t^{1 + 1} + 6(- 2) \times t + 10 \times 2 \times t + 60 \\ & = - 12t + 20t - 4t^{2} + 60 \\ & = (- 12 + 20) \times t - 4t^{2} + 60 \\ & = - 4t^{2} + 8t + 60
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\end{align*}
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\item
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\begin{align*}
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C & = (- 9x - 5)(3x + 4) \\ & = - 9x \times 3x - 9x \times 4 - 5 \times 3x - 5 \times 4 \\ & = - 9 \times 3 \times x^{1 + 1} + 4(- 9) \times x - 5 \times 3 \times x - 20 \\ & = - 36x - 15x - 27x^{2} - 20 \\ & = (- 36 - 15) \times x - 27x^{2} - 20 \\ & = - 27x^{2} - 51x - 20
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\end{align*}
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\item
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\begin{align*}
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D & = (6x - 2)(9x + 10) \\ & = 6x \times 9x + 6x \times 10 - 2 \times 9x - 2 \times 10 \\ & = 6 \times 9 \times x^{1 + 1} + 10 \times 6 \times x - 2 \times 9 \times x - 20 \\ & = 60x - 18x + 54x^{2} - 20 \\ & = (60 - 18) \times x + 54x^{2} - 20 \\ & = 54x^{2} + 42x - 20
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\end{align*}
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\item
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\begin{align*}
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E & = (- 8x - 4)^{2} \\ & = (- 8x - 4)(- 8x - 4) \\ & = - 8x \times - 8x - 8x(- 4) - 4 \times - 8x - 4(- 4) \\ & = - 8(- 8) \times x^{1 + 1} - 4(- 8) \times x - 4(- 8) \times x + 16 \\ & = 32x + 32x + 64x^{2} + 16 \\ & = (32 + 32) \times x + 64x^{2} + 16 \\ & = 64x^{2} + 64x + 16
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\end{align*}
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\item
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\begin{align*}
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F & = (- 8x - 10)^{2} \\ & = (- 8x - 10)(- 8x - 10) \\ & = - 8x \times - 8x - 8x(- 10) - 10 \times - 8x - 10(- 10) \\ & = - 8(- 8) \times x^{1 + 1} - 10(- 8) \times x - 10(- 8) \times x + 100 \\ & = 80x + 80x + 64x^{2} + 100 \\ & = (80 + 80) \times x + 64x^{2} + 100 \\ & = 64x^{2} + 160x + 100
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\end{align*}
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\item
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\begin{align*}
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G & = (- 10x + 6)^{2} \\ & = (- 10x + 6)(- 10x + 6) \\ & = - 10x \times - 10x - 10x \times 6 + 6 \times - 10x + 6 \times 6 \\ & = - 10(- 10) \times x^{1 + 1} + 6(- 10) \times x + 6(- 10) \times x + 36 \\ & = - 60x - 60x + 100x^{2} + 36 \\ & = (- 60 - 60) \times x + 100x^{2} + 36 \\ & = 100x^{2} - 120x + 36
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\end{align*}
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\item
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\begin{align*}
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H & = (\dfrac{- 6}{4} \times x - 7)^{2} \\ & = (\dfrac{- 6}{4} \times x - 7)(\dfrac{- 6}{4} \times x - 7) \\ & = \dfrac{- 6}{4} \times x \times \dfrac{- 6}{4} \times x + \dfrac{- 6}{4} \times x(- 7) - 7 \times \dfrac{- 6}{4} \times x - 7(- 7) \\ & = \dfrac{- 6}{4} \times \dfrac{- 6}{4} \times x^{1 + 1} - 7 \times \dfrac{- 6}{4} \times x - 7 \times \dfrac{- 6}{4} \times x + 49 \\ & = \dfrac{- 7(- 6)}{4} \times x + \dfrac{- 7(- 6)}{4} \times x + \dfrac{- 6(- 6)}{4 \times 4} \times x^{2} + 49 \\ & = \dfrac{42}{4} \times x + \dfrac{36}{16} \times x^{2} + \dfrac{42}{4} \times x + 49 \\ & = 49 + \dfrac{36}{16} \times x^{2} + \dfrac{42}{4} \times x + \dfrac{42}{4} \times x \\ & = 49 + \dfrac{36}{16} \times x^{2} + (\dfrac{42}{4} + \dfrac{42}{4}) \times x \\ & = 49 + \dfrac{36}{16} \times x^{2} + \dfrac{42 + 42}{4} \times x \\ & = \dfrac{36}{16} \times x^{2} + \dfrac{84}{4} \times x + 49
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\end{align*}
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\end{enumerate}
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\end{multicols}
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\end{solution}
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