# 2017 AMC 12A Problems/Problem 21

## Problem

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?

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

## Solution 1

At first, $S=\{0,10\}$.

$10x+10$ has root $x=-1$, so now $S=\{-1,0,10\}$.

$-x^{10}-x^9-x^8-x^7-x^6-x^5-x^4-x^3-x^2-x+10$ has root $x=1$, so now $S=\{-1,0,1,10\}$.

$x+10$ has root $x=-10$, so now $S=\{-10,-1,0,1,10\}$.

$x^4-x^2-x+10$ has root $x=2$, so now $S=\{-10,-1,0,1,2,10\}$.

$x^4-x^2+x+10$ has root $x=-2$, so now $S=\{-10,-2,-1,0,1,2,10\}$.

$2x-10$ has root $x=5$, so now $S=\{-10,-2,-1,0,1,2,5,10\}$.

$2x+10$ has root $x=-5$, so now $S=\{-10,-5,-2,-1,0,1,2,5,10\}$.

At this point, no more elements can be added to $S$. To see this, let

$a_{n}x^n + a_{n-1}x^{n-1} + ... + a_{2}x^2 + a_{1}x + a_0$ = $x(a_{n}x^{n-1} + a_{n-1}x^{n-2} + ... + a_{2}x + a_{1}) + a_0 = 0$

with each $a_i$ in $S$.

Since $x$ and the parenthesized term are both integers, $x$ must be a factor of $a_0$. However, $a_0$ is in $S$ and every number in $S$ already has all of its factors in $S$. Therefore, $x$ must be in $S$ and $S$ cannot be expanded. $\{-10,-5,-2,-1,0,1,2,5,10\}$ has $9$ elements $\to \boxed{\textbf{(D)}}$