a^2 = (x+y)^2 = x^2 + y^2 + 2xy = b + 2xy
xy = (a^2-b)/2

Du coup, a^3 = (x+y)^3 = x^3 + y^3 + 3xy^2 + 3x^2y
= x^3 + y^3 +3xy(x+y)

x^3 + y^3 = a^3 -3a(a^2-b)/2

Ou encore
x^3 + y^3 = a(3b-a^2)/2

Autre méthode

x+y = a
xy = (a^2-b)/2

On résoud
X^2-aX+(a^2-b)/2 = 0
Delta = a^2-2a^2+2b = 2b-a^2

x = (a+sqrt(2b-a^2)) / 2
y = (a-sqrt(2b-a^2)) / 2
(ou l'inverse)

Partant de là, on peut trouver tous les x^n + y^n
x^3+y^3 = (a^3+3a^2sqrt(2b-a^2)+3a(2b-a^2)+(2b-a^2)sqrt(2b-a^2) + a^3-3a^2sqrt(2b-a^2)+3a(2b-a^2)-(2b-a^2)sqrt(2b-a^2))/8
= (a^3+3a(2b-a^2))/4 = (6ab-2a^3)/4 = a(3b-2a^2)/2