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^2 + b^2 + 2ab = 1
2 + 2ab = 1
2ab = -1
ab = -1/2
a^3 + b^3 = (a + b) ( a^2 + b^2 -ab) = (1) (2 - -1/2) = 2 + 1/2 = 5/2
+769 
