+820 Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$
+37069 a+b = 1 square both sides
(a+b)^2 = 1
a^2 + b^2 + 2 ab = 1 but we are given a^2 + b^2 = 2 substitute this in
2 + 2 ab = 1
ab = -1/2
Now
(a+b)^3 = a ^3 + b^3 + 3 a b^2 + 3 a^2 b =
a^3 + b^3 + 3ab (a+b) Substitute in ab = -1/2 and a+b = 1
a^3 + b^3 + 3 (-1/2) (1)
(a+b)^3 = a^3 + b^3 - 3/2 if a+b = 1 then (a+b)^3 = 1 susttiute this in on the left
1 = a^3 + b^3 - 3/2 add 3/2 to both sides
5/2 = a^3 + b^3
