Example 6 Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta$ and $\alpha \beta$, find the sum of the coefficients of this polynomial.
(2005 Western Olympiad Problem)
Solve for the sum of coefficients in the expansion of $\alpha^{k}+\beta^{k}$, given $\alpha+\beta=1, \alpha \beta=1$, denoted as $S_{k}$. Then $\alpha, \beta$ are the roots of the equation $x^{2}-x+1=0$. Solving this, we get
$$
\alpha=\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}, \beta=\cos \frac{\pi}{3}-i \sin \frac{\pi}{3},
$$

Thus, $\alpha^{k}+\beta^{k}=\left(\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}\right)^{k}+\left(\cos \frac{\pi}{3}-\mathrm{i} \sin \frac{\pi}{3}\right)^{k}$
$$
\begin{array}{l}
=\left(\cos \frac{k \pi}{3}+\mathrm{i} \sin \frac{k \pi}{3}\right)+\left(\cos \frac{k \pi}{3}-\mathrm{i} \sin \frac{k \pi}{3}\right) \\
=2 \cos \frac{k \pi}{3} .
\end{array}
$$

Taking $k=2005$, we get $S_{k}=1$.