2018 AMC 12A Problems/Problem 21

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Problem

Which of the following polynomials has the greatest real root? $\textbf{(A) }   x^{19}+2018x^{11}+1   \qquad        \textbf{(B) }   x^{17}+2018x^{11}+1   \qquad    \textbf{(C) }   x^{19}+2018x^{13}+1   \qquad   \textbf{(D) }  x^{17}+2018x^{13}+1 \qquad  \textbf{(E) }   2019x+2018$

Solution

We can see that our real solution has to lie in the open interval $(-1,0)$. From there, note that $x^a < x^b$ if a, b are odd positive integers so $a<b$, so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). Finally, we can see that plugging in the root of $2019x+2018$ into B gives a negative, and so the answer is $\fbox{B}$. (cpma213)

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