Explications étape par étape:
1) • (1+√2)² = 1²+2√2 + √2² = 1 +2√2+2= 3+2√2
• (1+√2)³ = 1³+3√2+3√2²+√2³= 1+3√2+6+2√2
= 7+5√2
2) P(1+√2) = -3 (1+√2)³+4(1+√2)²+5(1+√2)-2
= -3(7+5√2) + 4(3+2√2) + 5(1+√2) -2
= -21 -15√2 + 12 + 8√2 + 5 +5√2 -2
= -6 -3√2
3) P(2) = -3(2)³+4(2)²+5(2)-2
= -24 + 16 + 10 -2
= 0
P(2)=0 donc 2 est une racine de P(x)
4)on a : P(x) = -3x³ + 4x² + 5x -2 = -3x³+6x²-2x²+4x+x-2
= -3x²(x-2) - 2x(x-2) + ( x-2)
= (x-2) (-3x²-2x+1)
donc Q(x) = -3x²- 2x + 1
5) Q(-1) = -3(-1)² -2(-1) + 1 = -3 + 2 + 1 = 0
donc -1 est un racine de Q(x) donc Q(x) est divisible par (x+1)
6) on a : Q(x) = -3x² - 2x + 1 = -3x² - 3x + x + 1
= -3x(x+1)+(x+1) =(x+1)(-3x+1)
7)on a : P(x) = (x-2)Q(x) =(x-2)(-3x²-2x+1)
P(x) = (x-2)(x+1)(-3x+1)
8) • A(x) = P(x) - (5x-4) Q(x)
= (x-2)(x+1)(-3x+1) - (5x-4)(x+1)(-3x+1)
= (x+1)(-3x+1) [(x-2)-(5x-4)]
= (x+1)(-3x+1)(-4x+2) = (-4x+2)(x+1)(-3x+1)
• A(x) = 0
(-4x+2)(x+1)(-3x+1) = 0
(-4x+2) = 0 ou (x+1) = 0 ou (-3x+1) = 0
-4x = -2 ou x = -1 ou -3x = -1
x = 1/2 ou x = -1 ou x = 1/3
donc S={-1; 1/3; 1/2}
