Réponse :
Explications étape par étape :
Bonsoir
on considère l'expression : A (x) = (x + 1) (2-x) -2 (x + 1) (2x + 3)
1. Développer A(x)
A (x) = (x + 1) (2-x) -2 (x + 1) (2x + 3)
A(x) = 2x - x² + 2 - x - 2 (2x² + 3x + 2x + 3)
A(x) = - x² + x + 2 - 2( 2x² + 5x + 3)
A(x) = - x² + x + 2 - 4x² - 10x - 6
A(x) = - 5x² - 9x - 4
et factoriser
A (x) = (x + 1) (2-x) -2 (x + 1) (2x + 3)
A(x) = (x + 1) (2 - x - 2(2x + 3))
A(x) = (x + 1) ( 2 - x - 4x - 6)
A(x) = (x + 1) (- 5x - 4)
Le facteur commun est ici souligné, on le met devant et on met le reste derrière
2. Calculer A(2), A(-1) et A(-2/3)
A(2) =(2 + 1) (- 5(2) - 4)
A(2) = 3 (- 10 - 4)
A(2) = 3 (- 14)
A(2) = - 42
A(- 1) =((-1) + 1) (- 5(-1) - 4)
A(- 1) = 0 (- 10 - 4)
A(- 1) = 0
A( - 2/3) =(- 2/3 + 1) (- 5(- 2/3) - 4)
A(- 2/3) = (- 2/3 + 3/3) (10/3 - 12/3)
A(- 2/3) = 1/3 ( - 2/3)
A(- 2/3) = (- 2/9)
3. Résoudre les équations suivantes:
a) A(x)= 0
A(x) =(x + 1) (- 5x - 4) = 0
soit x + 1 = 0 ou - 5x - 4 = 0
soit x = - 1 ou - 5x = 4
soit x = - 1 ou x = - 4/5
S = {- 1; - 4/5}
b) A(x) = - 4
A(x) = - 5x² - 9x - 4 = - 4
A(x) = - 5x² - 9x - 4 + 4 = 0
A(x) = - 5x² - 9x = 0
A(x) = - x (5x + 9) = 0
soit - x = 0 ou 5x + 9 = 0
soit x = 0 ou 5x = - 9
soit x = 0 ou x = - 9/5
S = { - 9/5; 0}
c) A(x)= -9x-4
A(x) = - 5x² - 9x - 4= - 9x - 4
A(x) = - 5x² - 9x - 4 + 9x + 4 = 0
A(x) = - 5x² = 0
- 5x² = 0
x² = 0
x = 0
S = {0}
d) A(x)= -5x²
A(x) = - 5x² - 9x - 4 = - 5x²
A(x) = - 5x² - 9x - 4 + 5x² = 0
A(x) = - 9x - 4 = 0
- 9x - 4 = 0
- 9x = 4
x = - 4/9
S = {- 4/9}
e) -9x-9
A(x) = - 5x² - 9x - 4 = - 9x - 9
A(x) = - 5x² - 9x - 4 + 9x + 9 = 0
A(x) = - 5x² + 5 = 0
A(x) = - 5 ( x² - 1) = 0 car (x² - 1) est de la forme a² - b² = (a +b ) (a - b)
- 5 (x + 1) (x - 1) = 0
soit - 5 (x+ 1) (x - 1) = 0
soit - 5 (x + 1) = 0 ou x - 1 = 0
soit x + 1 = 0 ou x - 1 = 0
soit x = - 1 ou x = 1
S = { - 1; 1 }
