When you leave the comfort of the real numbers, you might be mistaken about what a polynomial is.
Suppose you’re looking at polynomials over some finite field. Why would you do that? Numerous practical applications, but that’s a topic for another post.
You look in some reference that’s talking about polynomials and you see things like
x³ + x
referred to as a polynomial over the field with two elements, 0 and 1. “Yeah, I get it. It’s just like middle school, except the coefficients are zeros and ones.”
No, it’s not like middle school.
What happens when you stick in x = 0? You get 0. What happens when you stick in 1? You get 0, because 1 + 1 = 0 mod 2.
The function that takes a bit and returns the cube of the bit plus the bit itself always evaluates to 0 when working mod 2.
We have a non-zero polynomial that when interpreted as a function equals the zero function.
The resolution to this tension is that in general, a polynomial is not a function. You can associate a function with a polynomial by evaluating the polynomial for some value of x, but two different polynomials may correspond to the same function.
I have a vague memory of some professor correcting me for conflating a polynomial with a function. That seemed like pure pedantry at the time. It would have been helpful if said professor had explained why the distinction mattered. It never mattered in any example I could think of.
When you go back and look at how polynomials are defined over an arbitrary field, there are two possibilities. A reference may use powers of x and coefficients from the field, but say somewhere that the x is a “formal symbol” or something like that. The other possibility is to define polynomials as infinite sequences of field elements for which only a finite number of elements are nonzero, along with addition and multiplication rules that look just like what you’d expect for polynomials.
This may seem like unnecessary formalism. And for polynomials over real numbers it is. If two real polynomials differ as polynomials, i.e. they differ in one of their coefficients, then there is some place where they differ as functions. But that’s not true in general.