+1234 Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$
+1926 First, let's take a look at the first equation we are given.
\(a+b=1 \)
Squaring both sides, we have
\(a^2 + 2ab + b^2 = 1 \)
Now, we already know a^2 + b^2 is 2, so plugging that in, we have
\(2+2ab = 1\\ 2ab=-1\\ ab=-1/2\)
ab will come into handy later.
Now, let's notcie something real quick. We have
\(a^3 + b^3 = (a + b) ( a^2 + b^2 - ab)\)
We already know all the terms needed now! We have
\(a^3 + b^3 = (1) ( 2 +1/2)\\ a^3+b^3 = 5/2\)
So our answer is 5/2.
Thanks! :)
