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avatar+1636 

Let $a$ and $b$ be complex numbers. If $a + b = 4$ and $a^2 + b^2 = 6,$ then what is $a^3 + b^3?$

 Jun 9, 2024
 #1
avatar+1644 
-2

a^3 + b^3 = 56.

 Jun 9, 2024
 #2
avatar+759 
+1

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! :)

 Jun 9, 2024

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