If x + y = 1 and x^2 + y^2 = 2, then find x^3 + y^3.
x + y = 1 square both sides
x^2 + y^2 + 2xy = 1
2 + 2xy = 1
2 -1 = -2xy
1 = -2xy
-(1/2) = xy
1/2 = -xy
x^3 + y^3 factors as a sum of cubes = (x + y) ( x^2 - xy + y^2) =
( x + y) ( x^2 + y^2 - xy ) =
(1) ( 2 + 1/2) =
5/2
