Write an equation expressing c explicitly in terms of a and b if a = x + y, b = x^2 + y^2, c = x^3 + y^3.
a = x + y (1)
b = x^2 + y^2 (2)
c = x^3 + y^3 = ( x + y) (x^2 - xy + y^2) = ( x + y) (x^2 + y^2 - xy ) (3)
Square both sides of (1) and we get that
a^2 = x^2 + 2xy + y^2
-2xy = x^2 + y^2 - a^2
-2xy = b - a^2
-xy = [ b - a^2 ] / 2 (4)
Sub (1), (2) (4) into (3) and we have that
c = ( a) ( b + [ b - a^2] / 2 )
c = (a) ( b + b/2 - a^2/2 )
c = (a) ( (3/2)b - (1/2)a^2 )
c = (3/2)ab - (1/2)a^3