10.5. Find a polynomial with integer coefficients for which the number $\sqrt[3]{2}+\sqrt[3]{3}$ is a root.
10.5. Simple calculations show that

\[
\begin{gathered}
(\sqrt[3]{2}+\sqrt[3]{3})^{3}=5+3 \alpha \\
(\sqrt[3]{2}+\sqrt[3]{3})^{6}=133+30 \alpha+9 \beta \\
(\sqrt[3]{2}+\sqrt[3]{3})^{9}=2555+711 \alpha+135 \beta
\end{gathered}
\]

where \(\alpha=\sqrt[3]{6}(\sqrt[3]{2}+\sqrt[3]{3})\) and \(\beta=\sqrt[3]{36}(\sqrt[3]{4}+\sqrt[3]{9})\). Therefore, \(x^{9}-15 x^{6}-87 x^{3}-125=0\) for \(x=\sqrt[3]{2}+\sqrt[3]{3}\).