8. Let the quartic polynomial with integer coefficients $f(x)$ satisfy:
$$
f(1+\sqrt[3]{3})=1+\sqrt[3]{3}, f(1+\sqrt{3})=7+\sqrt{3} \text {. }
$$

Then $f(x)=$ $\qquad$
$\begin{array}{l}\text { 8. } x^{4}-3 x^{3}+3 x^{2}-3 x \text {. } \\ \text { Let } g(x)=f(x)-x, h(x)=g(x+1) \text {. } \\ \text { Given } h(\sqrt[3]{3})=g(1+\sqrt[3]{3})=0 \\ \Rightarrow h(x)=\left(x^{3}-3\right)(a x+b)(a, b \in \mathbf{Z}) \\ \Rightarrow h(\sqrt{3})=g(\sqrt{3}+1) \\ =f(1+\sqrt{3})-(1+\sqrt{3})=6 \\ \Rightarrow(3 \sqrt{3}-3)(a \sqrt{3}+b)=6 \\ \Rightarrow a=b=1 \\ \Rightarrow f(x)=g(x)+x=h(x-1)+x \\ =x^{4}-3 x^{3}+3 x^{2}-3 x .\end{array}$