TITLE: Why can't polynomials have negative exponents or division by a variable
QUESTION [31 upvotes]: Why can't:
$$2x^{-3} - 3x$$
or
$$\frac{1}{2x}$$
be polynomials too?
Why have a definition that excludes these algebraic forms?

REPLY [4 votes]: Here’s my understanding of the reason for the restrictiveness of the definition of “polynomial”, in response to @QuoraFea’s suggestion.
I’ll try to make my explanation as elementary as I can, with parenthetical expansions for the more technically inclined.
The set (ring, actually) of polynomials with real coefficients (more generally with coefficients in any commutative ring) has a “universal” property that the larger sets do not. Just to speak only of polynomials in one variable, the set of all such, $\mathbb R[x]$ ($R[x]$ for a general ring), has the property that the variable $x$ may be evaluated to any real number $a$ (to any element $\alpha$ of an algebra $A$ over the base ring $R$) so that this “evaluation mapping” can be applied to any polynomial at all (to any element of $R[x]$ at all). That is, once you’ve chosen $a$, then $f(a)$ makes sense as a real number (as an element of $A$) no matter what polynomial $\,f$ you look at.
Furthermore, the choice of $a$, once it’s done, gives you a function from polynomials to constants, I’ll call it $e_a$, namely $e_a(f)=f(a)$. And this function is both additive and multiplicative (it’s a ring morphism from $R[x]$ to $A$). That is, it satisfies $e_a(f+g)=e_a(f)+e_a(g)$ and $e_a(fg)=e_a(f)e_a(g)$, as well as $e_a(\mathbf1)=1$, where the bold-face $\mathbf1$ is the constant polynomial $1$, and the other $1$ is the ordinary unit element of $\mathbb R$. The function $e_a$ may be called the evaluation morphism. I stress that the domain is the set of polynomials, and the target space, also known as codomain, is $\mathbb R$ (the algebra $A$ in the general case).
There are, indeed, other algebraic structures that allow the evaluation functions to constants [for instance the set of all entire analytic functions on the line], but the ring of real polynomials $R[x]$ is the only one that allows the variable $x$ to be evaluated to an element of any larger structure containing $\mathbb R$ (to any $R$-algebra).