Réponse :
Bonjour
1) (x + 1)³ + 2(x² - 2x - 3) = (x + 1)(x + 1)² + 2x² - 4x - 6
= (x + 1)(x² + 2x + 1) + 2x² - 4x - 6
= x³ + 2x² + x + x² + 2x + 1 + 2x² - 4x - 6
= x³ + 5x² - x -5
= A(x)
2) B(x) = x² - 2x - 3 = (x + 1)(x - 3)
3) A(x) = (x+ 1)³ + 2(x + 1)(x - 3)
A(x) = (x + 1)[(x + 1)² + 2(x - 3)]
A(x) = (x + 1)(x² + 2x + 1 + 2x - 6)
A(x) = (x + 1)(x² + 4x - 5)
A(x) = 0 ⇔ x + 1 = 0 ou x² + 4x - 5 = 0
⇔ x = - 1 ou x² + 4x - 5 = 0
Résolvons l'équation x²+ 4x - 5 = 0
Δ = 4² - 4×1×(-5) = 16 + 20 = 36
x₁ =( -4 - 6)/2 =-5 et x₂ = (-4 + 6)/2 = 1
Au final, pour l'équation (E) on a S = {-5 ; -1 ; 1}
