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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

Key Flex
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    It can't be written that way, since the sum you've written doesn't converge. Also just because ten can be written as not a polynomial wouldn't mean that ten isn't still an integer and thus a constant polynomial with integer coefficients. – jgon Dec 27 '18 at 03:53
  • The series doesn't actually converge. The two aren't equal. – Sort of Damocles Dec 27 '18 at 03:54
  • All are saying that series written wouldn't converge. But for 10x is a Polynomial we all know and it can be written as 5x+5x and 5x+5x is also a polynomial and it also doesn't converge. Isn't this analogous to my question –  Dec 27 '18 at 04:01
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    There are no non finite series involved in defining polynomials. Where is this coming from? – copper.hat Dec 27 '18 at 04:02
  • $5x+5x$ isn't a good analogy for what you are asking... $5x+5x$ is a sum with two things in it... that's far simpler a notion than an infinite sum... The infinite sum $2-2+2-2+2+ \dots $ has some more details to be examined. You can find some details here. – Mason Dec 27 '18 at 04:04
  • 10 cannot be written as $ 10+(2-2+2-2+2-2+2-2..........)$ since the expression $2-2+2-2+2-2+2-2..........$ is meaningless. You must properly define everything or you end up with vague statements that lead to confusion without having a logical basis to build on. – CyclotomicField Dec 27 '18 at 04:06
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    "5x+5x is also a polynomial and it also doesn't converge" What? – Noah Schweber Dec 27 '18 at 04:06
  • Ya 5x+5x is a polynomial and it doesn't converge and 10+(2-2+2-2.....) Is not converging and thus be a polynomial, but why not –  Dec 27 '18 at 04:09
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    @user629353 "5x+5x is a polynomial and it doesn't converge" I don't think you know what "converge" means. – Noah Schweber Dec 27 '18 at 04:10
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    $$10=0x^3+0x^2+0x+10$$ $10$ is a polynomial – clathratus Dec 27 '18 at 04:26
  • Are you asking about the decimal representation of $10$? In which case $10=1\cdot 10^1 + 0\cdot 10^0$ which looks like a polynomial to me. – timtfj Dec 27 '18 at 04:38

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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.

Noah Schweber
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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.

littleO
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