Bonjour
1) P(-2) = 2*(-2)^3 - 3*(-2)^2 - 11*(-2) + 6 = -16 - 12 + 22 + 6 = 0
-2 est une racine de P
Donc P(x) est divisible par x - (-2) = x + 2
2) P(x) = (x + 2)Q(x)
Q(x) = ax² + bx + c avec a,b,c réels et a non nul
P(x) = (x + 2)(ax² + bx + c) = ax^3 + bx² + cx + 2ax² + 2bx + 2c
P(x) = ax^3 + (b + 2a)x² + (c + 2b) + 2c = 2x^3 - 3x² - 11x + 6
Par identification :
a = 2
b + 2a = -3
c + 2b = -11
2c = 6
a = 2
b = -3 - 2a = - 3 - 2(2) = -7
c = -11 - 2b = -11 -2(-7) = 3
c = 6/2 = 3
Q(x) = 2x² - 7x + 3
3)a) Q(x) = 0
2x² - 7x + 3 = 0
delta = b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25 > 0
x1 = [tex]\frac{-b-\sqrt{delta} }{2a}[/tex] = (7 - 5)/4 = 1/2
x2 = [tex]\frac{-b+\sqrt{delta} }{2a}[/tex]= (7 + 5)/4 = 3
S = { 1/2 ; 3 }
b) P(x) = 0
S = { -2 ; 1/2 ; 3 } car P(x) = (x + 2)Q(x)
c) Q(x) = a(x - x1)(x - x2) = 2(x - 1/2)(x - 3) = (2x - 1)(x - 3)
P(x) = (x + 2)Q(x) = (2x - 1)(x + 2)(x - 3)
d) 5P(x) - (2x - 3)Q(x) = 0
5(2x - 1)(x + 2)(x - 3) - (2x - 3)(2x - 1)(x - 3) = 0
(2x - 1)(x - 3)(5(x + 2) - (2x - 3)) = 0
(2x - 1)(x - 3)(3x + 13) = 0
2x - 1 = 0 ou x - 3 = 0 ou 3x + 13 = 0
x = 1/2 ou x = 3 ou x = -13/3
S = { -13/3 ; 1/2 ; 3 }
4) a) | x - 1 | < 1/3
-1/3 < x - 1 < 1/3
-1/3 + 1 < x < 1/3 + 1
2/3 < x < 4/3
b) 2/3 < x < 4/3
4/3 < 2x < 8/3
4/3 - 1 < 2x - 1 < 8/3 - 1
1/3 < 2x - 1 < 5/3
2/3 + 2 < x + 2 < 4/3 + 2
8/3 < x + 2 < 10/3
2/3 - 3 < x - 3 < 4/3 - 3
-7/3 < x - 3 < -5/3
c) P(x) = (2x - 1)(x + 2)(x - 3) = (-1) * (2x - 1)(x + 2=(-(x - 3))
5/3 < -(x - 3) < 7/3
(1/3)(8/3)(5/3) < - P(x) < (5/3)(10/3)(7/3)
40/27 < - P(x) < 350 / 27
-350 / 27 < P(x) < -40 / 27
