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 $S$, and where $a_n \ne 0$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?
This is another AMC question... and the problem is, I don't quite understand it. So when $S = \{0,10\}$, what does the polynomial look like? Is it $0x^2+10x$ because there are two numbers in the set? But then how would I account for $a_0$? I'm really confused, and any help would really be appreciated!
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