MathDB
Set Construction

Source: AMC 12 2017A/21

February 8, 2017
algebrapolynomialAMCAMC 122017 AMC 12A

Problem Statement

A set SS is constructed as follows. To begin, S={0,10}S=\{0,10\}. Repeatedly, as long as possible, if xx is an integer root of some polynomial anxn+an1xn1++a1x+a0a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 for some n1n\geq 1, all of whose coefficients aia_i are elements of SS, then xx is put into SS. When no more elements can be added to SS, how many elements does SS have?
<spanclass=latexbold>(A)</span>4<spanclass=latexbold>(B)</span>5<spanclass=latexbold>(C)</span>7<spanclass=latexbold>(D)</span>9<spanclass=latexbold>(E)</span>11<span class='latex-bold'>(A) </span> 4 \qquad <span class='latex-bold'>(B) </span> 5 \qquad <span class='latex-bold'>(C) </span> 7 \qquad <span class='latex-bold'>(D) </span> 9 \qquad <span class='latex-bold'>(E) </span> 11