@XanderHenderson That definition of a polynomial is an example of the type of definition you're criticising (unmotivated and based on formal expressions/pattern matching). Moreover you'd first need to define "formal variable".

@MichaelBächtold The definition is entirely motivated in a precalculus class, though not in that language.

In a standard precalculus class, it is usual to introduce functions, then start exploring different varieties of functions.

It turns out that there is an algebraic structure on the set of functions which allows us to add, multiply, and compose functions.

Polynomials are the set of functions which can be produced by adding an multiplying monomial functions, i.e. it is the set of functions "generated by" the identity function and a constant function.

So the definition of a polynomial that I usually give to students is "A polynomial is any function which can be obtained by adding and/or multiplying primitive monomial functions, where a primitive monomial function is a function of the form $x \mapsto x^n$, where $n\in\mathbb{N}\cup \{0\}$, and $x$ is a real variable."

In more technical language, a polynomial is a linear combination of monomials.

Which is to say, it is a sum of the form $\sum_{j=0}^{n} a_j x^j$.