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

 Feb 23, 2024
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\(\begin{cases} a + b = 4\\ {a}^{2} b + {b}^{2} a = 6 \end{cases}\)

\(\begin{cases} a + b = 4\\ ab(a+b) = 6 \end{cases}\)

\(\begin{cases} a+b=4 \\ ab=\frac{3}{2}\end{cases}\)

\({a}^{3}+{b}^{3}=(a+b)({(a+b)}^{2}-3ab)\)

\({a}^{3}+{b}^{3}=4(16-3*\frac{3}{2})=46\)

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 Feb 23, 2024

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