Réponse :
EX1 Factoriser les expressions suivantes
A = 4ac - 8ac = 4ac(1 - 2) = - 4ac
B = (- x + 2)(6 x - 5) + ( 2 - x) = (2 - x)(6 x - 5 + 1) = (2 - x)(6 x - 4) = 2(2- x)(3 x -2)
C = 4 x² + 9 - 12 x ⇔ C = 4 x² - 12 x + 9 identité remarquable a²-2ab+b² = (a-b)²
C = 4 x² - 12 x + 9 = (2 x - 3)²
f(x) = - x² - 2 x - 1 ⇔ f(x) = - (x² + 2 x + 1) = - (x + 1)² IR a²+2ab+b² = (a+b)²
g(x) = 15 - 4 x² ⇔ g(x) = √15² - (2 x)² IR a²-b² = (a+b)(a-b)
g(x) = (√15 - 2 x)(√15+2 x)
h(x) = (7 x - 26)(11 x + 8) - (12 x + 4)(7 x - 26) = (7 x - 26)(11 x + 8 - 12 x - 4)
= (7 x - 26)(4 - x)
k(x) = 36 - (3 - 7 x)² ⇔ k(x) = 6² - (3 - 7 x)² IR a²-b² = (a-b)(a+b)
k(x) = (6 - 3 + 7 x)(6 + 3 - 7 x) = (7 x + 3)(9 - 7 x)
l(x) = (x-7)² - (3 x - 10)² IR idem que ci-dessus
l(x) = (x - 7 + 3 x - 10)(x-7 - 3 x + 10) = (4 x - 17)(3 - 2 x)
EX2 Etudier le signe des fonctions
f(x) = (4 - x)(3 x + 7)
x - ∞ - 7/3 4 + ∞
4 - x + + 0 -
3 x + 7 - 0 + +
f(x) - 0 + 0 +
g(x) = (5 x - 65)(2 - 7 x)
x - ∞ 2/7 13 + ∞
5 x - 65 - - 0 +
2 - 7 x + 0 - -
g(x) - 0 + 0 -
h(x) = x²(4 x - 7)(x + 3)
x - ∞ - 3 0 7/4 +∞
x² + + 0 + +
4 x - 7 - - - 0 +
x + 3 - 0 + + +
h(x) + 0 - 0 - 0 +
l(x)
x - ∞ 0 4/3 + ∞
l(x) + 0 - 0 +
Explications étape par étape
