Is 10 a polynomial?

Question

How 10 is a constant polynomial since it can be written as $10+(2-2+2-2+2-2+2-2..........)$ and thus having infinitely many terms. Also from Wikipedia's definition a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. My expression of 10 also satisfying the definition but contains infinite terms

Answer 1

There are lots of silly ways to write $10$ (ignoring the fact that what you've written doesn't really mean anything). For example, $$10=5-\sin(\pi)+3!+e^{i\pi}.$$ But none of this changes the fact that one of the ways to write $10$ is, well, "$10$" - and it's the fact that it can be written in such a way that makes it a polynomial. We don't care about the existence of other ways to write it. Similarly, an integer $a$ is even if $a$ can be written as $2\cdot b$ for some integer $b$; we can write $12$ as both $2\cdot 6$ and $3\cdot 5-1-2!$, and the fact that the former works means that $12$ is even regardless of the silliness of the latter.

Answer 2

If we want to be very precise, we can use the precise definition of a polynomial. One definition is that a polynomial (with coefficients that are real numbers) is a sequence $(a_0, a_1, a_2, \ldots)$ of real numbers such that $a_k = 0$ for all sufficiently large integers $k$.

By this definition, the number 10 is technically not a polynomial. However, people will often use the symbol 10 to denote the polynomial $(10,0,0,\ldots)$. This is an example of a symbol being "overloaded", which happens sometimes in math. Hopefully the meaning will always be clear from the context.