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

 Apr 24, 2024
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We can use the fact (a+b)3=a3+3a2b+3ab2+b3 to our advantage.

 

Expanding the left side, we get

 

\begin{align*} (a + b)^3 &= a^3 + 3a^2 b + 3ab^2 + b^3 \ &= a^3 + b^3 + 3ab (a + b). \end{align*}

 

Since we are given that a+b=4, this becomes a^3 + b^3 + 3ab(4) = 6.

 

Then a^3 + b^3 = −10​.

 Apr 28, 2024

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