Gonzalez Franklin
03/12/2024 · High School
13 Factoriser chacune des expressions suivantes a ) \( F=x^{2}+x \) b ) \( G=2 a+3+2\left(\frac{3}{2}-a\right)(2 a+3) \) c) \( H=2 y(1-y)-(1-y) \) d) \( F^{\prime}=5 x^{3}-8 x^{2}-x \) e) \( G^{\prime}=(4 t-1)(2-t)+2-t-3(2-t) \)
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Step-by-step Solution
Voici la factorisation de chacune des expressions demandées :
a) Pour \( F = x^{2} + x \) :
\[
F = x(x + 1)
\]
b) Pour \( G = 2a + 3 + 2\left(\frac{3}{2} - a\right)(2a + 3) \) :
D'abord, simplifions l'expression :
\[
G = 2a + 3 + 2\left(\frac{3}{2} - a\right)(2a + 3)
\]
Calculons \( 2\left(\frac{3}{2} - a\right)(2a + 3) \) :
\[
= 2\left(\frac{3}{2}(2a + 3) - a(2a + 3)\right) = 2\left(3a + \frac{9}{2} - 2a^2 - 3a\right) = 2\left(-2a^2 + \frac{9}{2}\right) = -4a^2 + 9
\]
Donc,
\[
G = 2a + 3 - 4a^2 + 9 = -4a^2 + 2a + 12
\]
Factorisons :
\[
G = -2(2a^2 - a - 6) = -2(2a + 6)(a - 1) = -2(2(a + 3)(a - 1))
\]
c) Pour \( H = 2y(1 - y) - (1 - y) \) :
\[
H = (1 - y)(2y - 1)
\]
d) Pour \( F' = 5x^{3} - 8x^{2} - x \) :
\[
F' = x(5x^{2} - 8x - 1)
\]
Pour factoriser \( 5x^{2} - 8x - 1 \), nous cherchons les racines :
\[
x = \frac{-(-8) \pm \sqrt{(-8)^{2} - 4 \cdot 5 \cdot (-1)}}{2 \cdot 5} = \frac{8 \pm \sqrt{64 + 20}}{10} = \frac{8 \pm \sqrt{84}}{10} = \frac{8 \pm 2\sqrt{21}}{10} = \frac{4 \pm \sqrt{21}}{5}
\]
Donc,
\[
F' = x(5(x - \frac{4 + \sqrt{21}}{5})(x - \frac{4 - \sqrt{21}}{5}))
\]
e) Pour \( G' = (4t - 1)(2 - t) + 2 - t - 3(2 - t) \) :
D'abord, simplifions :
\[
G' = (4t - 1)(2 - t) + 2 - t - 6 + 3t = (4t - 1)(2 - t) - 4 + 2t
\]
Calculons \( (4t - 1)(2 - t) \) :
\[
= 8t - 4t^2 - 2 + t = -4t^2 + 9t - 2
\]
Donc,
\[
G' = -4t^2 + 9t - 2
\]
Pour factoriser, nous cherchons les racines :
\[
t = \frac{-9 \pm \sqrt{9^{2} - 4 \cdot (-4) \cdot (-2)}}{2 \cdot (-4)} = \frac{-9 \pm \sqrt{81 - 32}}{-8} = \frac{-9 \pm \sqrt{49}}{-8} = \frac{-9 \pm 7}{-8}
\]
Les racines sont :
\[
t_1 = \frac{-2}{-8} = \frac{1}{4}, \quad t_2 = \frac{-16}{-8} = 2
\]
Donc,
\[
G' = -4(t - \frac{1}{4})(t - 2)
\]
Voilà les factorisations des expressions demandées.
Quick Answer
a) \( F = x(x + 1) \)
b) \( G = -2(2a + 6)(a - 1) \)
c) \( H = (1 - y)(2y - 1) \)
d) \( F' = x(5(x - \frac{4 + \sqrt{21}}{5})(x - \frac{4 - \sqrt{21}}{5})) \)
e) \( G' = -4(t - \frac{1}{4})(t - 2) \)
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