2017 AMC 12A Problem 21

A set S is constructed as follows. To begin, S = \{0,10\}. Repeatedly, as long as possible, if x is an integer root of some polynomial a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{1}x + a_0 for some n\geq{1}, all of whose coefficients a_i are elements of S, then x is put into S. When no more elements can be added to S, how many elements does S have?

\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ 11

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.