+2653 Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$
+1926 Ok, we have to do a number of things to set up this equation.
First off, we know that \((a+b)^2=a^2+2ab+b^2\)
Plugging in the values we know, we get that
\(4^2=6+2ab\\ 10=2ab\\ ab=5\)
It is crucial for us to know this information.
Now, let's take a look at a^3+b^3.
We know that \((a+b)^3=a^{3}+3a^{2}b+3ab^{2}+b^{3}\)
Now, take a look at the middle two terms. We can factor them to get \((a+b)^3=a^{3}+3ab(a+b)+b^{3}\)
We already know all these terms! We get
\(4^3=3(5)(4) +a^3+b^3\\ 64-60=a^3+b^3\\ 4=a^3+b^3\)
So 4 is our answer!
Thanks! :)
