Bonjour,
1) (u+v)^3 = u^3 + 3u^2v + 3uv^2 + v^3 = (u^3+v^3) + 3uv(u+v) = 4 + 15(u+v)
Soit (u+v)^3 - 15(u+v) - 4 = 0
==> (u+v) est solution de(E)
2) X^2 - 4X + 125 = X^2 - (u^3+v^3)X + (uv)^3 = (X-u^3)(X-v^3)
==> u^3 et v^3 solutions de X^2 - 4X + 125
Delta = 16-500 < 0 ==> pas de solution
3) X^2 - 4X + 125 = 0
<=> (X-2)^2 -4 + 125 = 0
<=> (X-2)^2 = -121
<=> (X-2)^2 = 121i^2 = (11i)^2
==>X-2 = 11i ou X-2 = -11i
Soit X = 2 + 11i ou X = 2 - 11i
4) (2+i)^3 = 2^3 + 3x2^2xi + 3x2xi^2 + i^3 = 8 + 12i - 6 - i = 2 + 11i
idem (2-i)^3 = 2 - 11i
==> u = 2+11i et v = 2-11i
==> u+v = 4
==> (E) devient :
(x-4)(ax^2 + bx + c) = 0
on développe ...
ax^3 + (ab-4a)x^2 + (c-4b)x - 4c = 0
Par analogie des termes de même degré :
a=1 b=4 c=1
==> (E) <=> (x-4)(x^2 + 4x + 1) = 0
x^2 + 4x + 1 = 0
Delta = 12
==> x = -2 + racine(3) ou x = -2 - racine(3)
