Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$
a + b = 1 square both sides
(a + b)^2 = 1
a^2 + 2ab + b^2 =1
a^2 + b^2 + 2ab = 1
2 + 2ab = 1
2ab = -1
ab = -1/2
-ab = 1/2
a^3 + b^3 = ( a + b) ( a^2 + b^2 - ab)
= (1) ( 2 + 1/2)
= 5 / 2
+569 
