what are necessary and sufficient condition for expression to be polynomial expression

Question

What defines or construe as polynomial? Which expression can be called as polynomial expression?

Answer 1

From a purely algebraic point of view, a polynomial in the indeterminate $X$ with coefficients in a certain set $R$ (usually with a ring structure at least) is an expression of the form $$a_0+a_1X+a_2X^2+\cdots+a_nX^n=\sum_{k=0}^n a_k X^k,$$ where $n\in \mathbb N_0$ and $a_0,a_1,\ldots,a_n \in R$. The set of all such expressions is sometimes represented by $R[X]$.

You can also have polynomials in several variables and other generalizations.

But from another angle, what is essential to polynomials is that there's a finite number of terms. And the indeterminate $X$ could be ommited and in its place we could have writen an expression of the form $$(a_0,a_1,\ldots,a_n)$$ or maybe $$(a_0,a_1,\ldots,a_n,0,0,\ldots).$$

In this sense, the set $R[X]$ can also be thought of in an equivalent way as the set of all sequences in $R$ with just a finite number of non-zero terms.

I would say that nothing else is "essential" to polynomials, although in most situations they'd be uninteresting if we hadn't certain operations and actions defined in that set. The definitions are the usual ones, and can be written for any of both conventions of notation... but the formula with $X$ makes them —the product specially— seem much more natural (and consequently turns out to be easier for calculations.) That's one of the reasons why we most of the times you'll find them in the form with $X$.

---

Something else.

But if you're studying calculus, pre-calculus, mathematical analysis, etc., a polynomial is the expression of a certain functional relation $f$ (loosely speaking, it's a kind of function) between two sets $A$ and $B$ (usually subsets of $\mathbb R$ or $\mathbb C$, but the fundamental is that you have a 'sum' and 'product' in $A$), given by $$f \colon A \to B, \qquad f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n,$$ where $n\in \mathbb N_0$ and $a_0,a_1,\ldots,a_n \in A$.