Proving that a function is not a polynomial

Question: Prove that the function $\displaystyle f(x) = \frac{x}{x^2 + 1}$ is not a polynomial.

My attempt (for which I got no marks as "it's not a proof") was to define a polynomial function as $\displaystyle \frac{P(x)}{Q(x)}$ where $\displaystyle Q(x)$ is always equal to 1, and since $\displaystyle x^2 + 1$ does not equal to 1 for any given value of $\displaystyle x$, then it's not a polynomial.

So how does one approach the question? I was thinking of rewriting $\displaystyle f(x) = \frac{x}{x^2 + 1}$ as $\displaystyle f(x0) = \dfrac{1}{x + \frac{1}{x}} = (x + \frac{1}{x})^{-1}$ and then, since a polynomial function has a degree $\displaystyle n \geq 0$, and this function has a degree -1, it is by definition not a polynomial. But is that really a proof either? Not sure what to do.

Re: Proving that a function is not a polynomial

I think I would try assuming f(x) = P(x), where P(x) is a polynomial, and see if I could deduce a contradiction.

We would then have

$\displaystyle (x^2+1) P(x) = x$

Hmmm...