Which of the following polynomials has the greatest real root? $\textbf{(A) } x^{19}+2018x^{11}+1 \qquad$ $\textbf{(B) } x^{17}+2018x^{11}+1 \qquad$ $\textbf{(C) } x^{19}+2018x^{13}+1 \qquad$ $\textbf{(D) } x^{17}+2018x^{13}+1 \qquad$ $\textbf{(E) } 2019x+2018$
To 'find' real roots we can set each expression $A$ to $E$ equal to $0$, which makes it pretty clear that the value $x$ must be negative (for else they'd be huge positive numbers), but also pretty small, since the terms with variable $x$ in each expression (despite their coefficients) must equal $-1$, for (A) to (D). $x=-1$ or $x=0$ will not work, however, so we need $x$ to be $-1<x<0$.
For (E) we can just solve for $x$ from $2019x + 2018=0$, but that gives an $x$ pretty close to $-1$ and I need to see if there are any greater roots (i.e. closer to $0$).
However, now I'm stuck and don't know how to move on. Any help would really be appreciated!