6.3.1 ** Find an integer-coefficient polynomial $f(x)$ such that $\sqrt[2]{2}+\sqrt[3]{3}$ is one of its roots.



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Let $x=\sqrt[3]{2}+\sqrt[3]{3}$, then $x-\sqrt[3]{2}-\sqrt[3]{3}$, cubing both sides we get
$$
x^{3}-3 \sqrt[3]{2} x^{2}+3 \sqrt[3]{4} x-2=3, x^{3}-5=3 x \sqrt[3]{2}(x-\sqrt[3]{2})=3 x \sqrt[3]{6} \text {, }
$$

cubing again, we get
$$
x^{9}-15 x^{6}+75 x^{3}-125=162 x^{3} \text {. }
$$

Let $f(x)=x^{9}-15 x^{6}-87 x^{3}-125$, then $\sqrt[3]{2}+\sqrt[3]{3}$ is a root of $f(x)$.