When we say that number 5 can be counted as a polynomial, do we make some additional assumptions about values of $0^0$ or x? https://www.mathsisfun.com/algebra/polynomials.html
At the link above it's said that $5$ is a polynomial. Basically because we can imagine it to be $5x^0$.
But it seems for me, that in order for this to be true, we need to believe/assume one of two things:
Either $0^0 = 1$ or $x$ can't be equal to $0$.
Thus, if I have $2x^3 + 5$, then to count it as a polynomial (if $5$ isn't a polynomial, then neither is $2x^3 + 5$. By same reason $2x^3  + \#$ isn't a polynomial. In order to get a polynomial via addition, both parts must be polynomials) I must either define $0^0$ as $1$ or forbid $x$ from being equal $0$.
Am I correct?
 A: Definitions are necessary to keep our math rigorous, but the math comes before the definitions.
In this case, it is clear that both $5$ and $2x^3+5$ should be polynomials. So the definitions had better make them be polynomials. How the definitions do it can vary, because there are several ways to define a polynomial.
Even if we limit ourselves to real polynomials in just one variable, here are some options:

*

*We can say that a polynomial is an expression of the form $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x^1 + a_0$ for any nonnegative integer $n$, where $a_0, a_1, \dots, a_n$ are real numbers. With this definition, there is no worry about $0^0$.

*We can say that a polynomial is an expression of the form $\sum_{i=0}^n a_i x^i$ with the same rules about $n, a_0, a_1, \dots, a_n$. We have to say that $x^0 = 1$ for this definition to be equivallent to the first, but we haven't said anything about substituting values for $x$ yet, so there's still nothing to worry about.

*We can define polynomials recursively: $x$ is a polynomial, any real number is a polynomial, and any sum or product of polynomials is a polynomial. Here, $x^0$ also never shows up; constants are included separately.

*We can say that a polynomial is a function $\mathbb R \to \mathbb R$ with a formula given by one of the definitions above. Then if we pick definition 2 to use, we need to say that $0^0 = 1$.

Usually, we make a distinction between a "polynomial expression" (definitions 1-3) and a "polynomial function" (definition 4, using any of 1-3 "under the hood") when we're trying to be formal.
