I'm curious about the following problem. Given two polynomials $p,q:\mathbb{R}\rightarrow \mathbb{R}$ is it possible to determine automatically if $p(x)\leq q(x)$ for all $x\in\mathbb{R}$?

I assume, of course, that we are given the expressions for $p$ and $q$:
$p(x)=a\_n x^n + \dots a\_1 x + a\_0$ and $q(x)=b\_m x^m + \dots b\_1 x + b\_0$
with $m,n\in \mathbb{N}$.

If this is possible, is it a consequence of some general theorem that, perhaps, extends to other kinds of (commutative) abstract algebras such as, e.g., the algebra of continuous maps from $X$ compact Hausdorf to $\mathbb{R}$?

Thank you! 

PS: I guess the question boils down to: is it possible to verify if $q-p\geq 0$. Sorry if the question is trivial but I don't know if this is possible.