Let's say that we have a polynomial in the following form:
$$
f(x) = d_0 + d_1 \cdot x + d_2 \cdot x^2 + d_3 \cdot x^3 + d_4 \cdot x^4 + .... + d_n \cdot x^n
$$
We want to derive the representing Taylor series for this, so we need to determine the coefficients of the Taylor Series. Let's keep it simple and center it around zero:
$$
\begin{aligned}
f(x) & = \sum_{j=0}^\infty f^{(j)}(0) \cdot {{x^j} \over j!}\\
f(0) & = d_0\\
f^{(1)}(0) & = d_1\\
f^{(n)}(0) & = n! \cdot d_n\\
\end{aligned}
$$
Beyond n all derivatives are zero
$$
\begin{aligned}
\therefore f(x) & = \sum_{j=0}^n j! \cdot d_j \cdot {{x^j} \over j!}\\
& = \sum_{j=0}^n d_j \cdot x^j
\end{aligned}
$$
So the Taylor series of a polynomial is the polynomial itself.