Question

Which of the following polynomials has the greatest real root?

$x^{19} + 2018x^{11} + 1$

$x^{17} + 2018x^{11} + 1$

$x^{19} + 2018x^{13} + 1$

$x^{17} + 2018x^{13} + 1$

$2019x + 2018$

Solution:

Each polynomial in choices A–D has no positive root and exactly one negative root, which lies in $(-1, 0)$ (it is positive at $0$ and negative at $-1$ ) and is increasing there. On the interval $(-1, 0),$ $x^{19} \lt x^{17}$ and $x^{13} \lt x^{11}.$ Increasing a term makes the polynomial larger, which pushes its root to the left (smaller). So the smallest exponents give the greatest root, favoring choice B ( $x^{17} + 2018x^{11} + 1$ ) over A, C, and D.

The linear choice E has root $-\tfrac{2018}{2019} = -\left(1 - \tfrac{1}{2019}\right),$ very close to $-1;$ evaluating the polynomial of choice B there gives a negative value, so B's root lies to the right of E's. Hence B has the greatest real root.

Thus, the correct answer is B.

2018 AMC 12A Problem 21

Solution:

Problem 21 in Other Years