Suppose that $P(x) = a\_m x^m + \dots + a\_0$ and $Q(x) = b\_n x^n + \dots + b\_0$ are two polynomials, with $m > n > 1$ and $a\_m > b\_n > 0$. Suppose that $P$ has $m$ distinct real roots $y\_1<\dots<y\_m$ and $Q$ has $n$ distinct real roots $z\_1<\dots<z\_n$.


Is the following claim true: 
$P(x) - Q(x)$ is strictly increasing for all $x \geq \max \{y\_m, z\_n\}$.


Note: I have a missing step in a larger proof, which would be fixed by a result like this. In my particular application, I have a bit more specific polynomials and with every numerical example I can come up with the result seems to hold, but I think this result should be more general.



Comparing tails of polynomial functions
Try $P(x) = 2 x (x+1)(x+1/3)$ and $Q(x) = x(x+1)$. Note that $P'(0) - Q'(0) = -1/3$.


