In the following I'm going to call
-
a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication)
that has the form $a_{n}x^{n}+\cdots+a_{1}x+a_{0}$, where $x$ is
some fixed element of said structure, -
a polynomial function a function of the form $x\mapsto a_{n}x^{n}+\cdots+a_{1}x+a_{0}$ (where the same algebraic considerations as above apply)
-
a polynomial an element of the form $a_{n}X^{n}+\cdots+a_{1}X+a_{0}$ with indeterminates $X$ (these
can be formalized, if we are in a ring, with strings of $0$s and
a $1$).
Note that when we are very rigorous/formal, polynomial expressions
and polynomials are something different (although in daily life we
often use them synonymously). Polynomial functions and expressions
are also different from each other although in this case the relationship
is a closer one, since every polynomials expression can be interpreted
as a polynomial function evaluated at a certain point (thus "polynomial
functions" are something more general than "polynomial expressions").
My question is: Why do we use polynomials ? It seems to me that every
piece of mathematics I have encountered so far, one could replace
every occurrence of polynomials with polynomial expressions/functions
without any major change in the rest of the proof/theorem/definition
etc.
The only reasons that I can see to use polynomials are the following
two:$$ $$
1. After one makes the idea precise that one can plug "suitable"
elements into polynomials (which may lie a ring containing the ring
in which the coefficients live in), one can save time in certain setting,
by handling the "plugging of suitable elements into polynomials"
more elegantly: For example in case of the theorem of Cayley-Hamilton,
which in its "polynomial function" version would look like:
Let $A$ be an $n\times n$ matrix over $K$, whose characteristic
polynomial (function) is $x\mapsto a_{n}x^{n}+\cdots+a_{1}x+a_{0}$.
Then
$$
a_{n}A^{n}+\cdots+a_{1}A+a_{0}I=0.
$$
whereas the "polynomial" version looks more elegant:
Let $A$ be an $n\times n$ matrix over $K$, whose characteristic
polynomial is $p_{A}\in K\left[X\right]$. Then
$$
p_{A}\left(A\right)=0.
$$
2. The only thing that polynomials can "do", but algebraic expressions/functions can't, is to be different, when the algebraic expressions/functions are the same (i.e. there's a theorem that tells us that the mapping of polynomials to polynomials expressions/functions
isn't injective, if the field is finite). Maybe this small difference makes a big enough difference
to consider polynomials after all, but as I said, I haven't encountered
any situation in which this difference could manifest itself.
(I'm guessing that maybe cryptography or higher number theory really needs
polynomials and not just polynomial expressions/functions. Since I don't know
anything about these subjects, I would be very happy with an example
of a theorem (whose content isn't merely technical as it is the case with the theorem above) involving polynomials, where these absolutely cannot
be replaced by polynomial expressions/functions. Conversely I would also be happy with an authoritative statement from someone knowledgeable that indeed we could dispense of polynomials, if we wanted to.)
Best Answer
You write that polynomial functions are "more general" than polynomial expressions. In fact the exact opposite is the case: multiple polynomial expressions may be the same function (on a finite field), so it's polynomial expressions that are more general.
Here are a few ways that polynomial expressions are useful where it could (or would) be awkward to use polynomial functions.
If $F$ is a field of $p^r$ elements, for a prime $p$ and positive integer $r$, then $F$ is the splitting field over ${\mathbf F}_p$ of $x^{p^r} - x$. But for every $a \in F$, $a^{p^r} = a$, so $a^{p^r} - a = 0$. You don't want to say $F$ is the splitting field of, well, "zero". The distinction between the nonzero polynomial $x^{p^r} - x$ in $F[x]$ and the zero polynomial is essential for that.
In case you think "I don't care about finite fields", well, some properties of sequences of integers are proved by making them into the coefficients of a polynomial and then reducing the polynomial mod p, which places the polynomial into ${\mathbf F}_p[x]$, where there are convenient features that are not available in ${\mathbf Z}[x]$. One example of this is a slick proof of Lucas's congruence on binomial coefficients. It's crucial for this that we can recover the coefficients of a polynomial in ${\mathbf F}_p[x]$ even if its degree is greater than $p$, which is impossible if you think of polynomials in ${\mathbf F}_p[x]$ as functions ${\mathbf F}_p \rightarrow {\mathbf F}_p$.
Quotient rings of $A[x]$, where $A$ is a ring, may not look like functions from an elementary point of view. Do you feel it's easier to work with ${\mathbf Q}[x]/(x^5)$ using the viewpoint of polynomials as functions rather than expressions?
Would you like to define the ring of noncommutative polynomials in two variables as functions?
Generating functions are important objects in mathematics (not just in combinatorics), and they are formal power series rather than numerical series. Most of them have positive radius of convergence (assuming their coefficients are in a ring like ${\mathbf R}$ or ${\mathbf C}$ and thus have a region where it makes sense to substitute a scalar for the variable at all), so they can be treated as functions on the domain of convergence, but it is convenient that the theory of formal power series doesn't really need to rely on the radius of convergence to get off the ground. Sometimes a generating function may even have radius of convergence equal to 0 but still contain useful information that can be legally extracted because formal power series can be developed without needing a radius of convergence. Of course a lot of important intuition about the way formal power series are manipulated is inspired by the properties of classical numerical power series as functions.
Your question mentions 3 types of polynomial objects: polynomial expressions, polynomial functions, and polynomials as strings of coefficients. I'd say the first and third are really synonymous (the third is really just a rigorous form of the first, not materially different in practice), but they're not the same as polynomial functions if you work over a finite field. If the coefficients are an infinite field then different polynomial expressions do define different functions on the field.