Question

Is the domain of a polynomial always `RR` ?

Answer 1

Yes. See the explanation.

Explanation:

Domain of the function is the set of all values for which we can calculate function expression. That is always possible in the case of additive and multiplicative operations (e.g. we can always use `+` and `*`).
Polynomials are constructed with those operations, so we can calculate `P(x)` for `AAx in R` and hence `D_f=R`.

Answer 2

Any polynomial in one variable is defined on all of `RR` or `CC`. In Calculus we are mostly concerned with `RR` or `CC` so one of these would be the implicit domain.

Explanation:

Any polynomial expression in one variable is defined on all of `RR`. It is also defined on all of `CC`, `QQ`, `ZZ_7`, H,...

There's a good argument for being explicit about the domain of a function.

For example, suppose `f(x) = 1/x`. I think it's much better to say `f: (-oo,0) uu (0, oo) -> RR`, specifying the domain explicitly, rather than saying `f(x)` is a function (implicitly on Real numbers since the letter `x` is used rather than `z`) that is not defined when `x = 0`.

For completeness, I should mention that in the context of calculus the implicit domain of a polynomial in `n` variables is `RR^n` or `CC^n`

Answer 3

Mathematically, yes. In applications (word problems), not always.

Explanation:

In your example, `f(x) = (2x+1)^3`, considered simply as a mathematical function, has domain `RR`.

If the function has arisen in an application in which `x` represents a distance, the the domain would be restricted to `[0,oo)`. Or, perhaps to `(0, oo)` or to a bounded interval if we have a maximum as well.