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A polynomial expression can be written in this form $$a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x+a_0$$ Therefore, this is a polynomial $$5x^4+3x^3+4x^2+3x+2$$ I understand this fairly well, because $n=4$. I know that when $n=2$, it is still a polynomial. $$4x^2+3x+2$$ But if I attempt to use the formula for $n=2$, I will end up with something like this. $$4x^2+3x^1+4x^2+3x+2$$ Do you see my reasoning? I want to have three values of $a$ such as $a_0=2,a_1=3,a_2=4$.

The above form of the polynomial expression using the $\ldots$ operator appear to be requiring at least 4 $+$ symbols and the duplication of $a_1, a_2$ when $n=2$.

I do not completely understand the usage of the $\dots$ operator.

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    $\begingroup$ The sum notation $\,\sum_{k=0}^n a_kx^k\,$ is always unambiguous. The "$\ldots$" shorthand stands for the same sum, and is often used for convenience in less formal contexts. $\endgroup$ – dxiv Jul 21 '18 at 2:30
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    $\begingroup$ It is not an operator. It’s a way of suggesting the desired form without going into formalities. $\endgroup$ – Lubin Jul 21 '18 at 3:49
  • $\begingroup$ I knew about the sum notation, it's true that it does remove the ambiguity. Is there a reason to use the ellipsis over the sum? $\endgroup$ – Cedric Martens Jul 21 '18 at 11:22
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    $\begingroup$ I have to wonder if the OP really thinks that $$n+(n-1)+(n-2)+\cdots+2+1$$ means $2+1+0+2+1$ when $n=2$. $\endgroup$ – Martin Argerami Jul 21 '18 at 12:36
  • $\begingroup$ I know that's not the case, as I showed by I know that when $n=2$, it is still a polynomial. $$4x^2+3x+2$$ However, the notation did not seem to imply that terms must be removed unlike what the accepted answer pointed out. $\endgroup$ – Cedric Martens Jul 21 '18 at 12:46
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Ellipses ($\ldots$) are not an operator: they're a piece of informal mathematical notation, meaning "fill in the pattern in the obvious way." As you've observed, in some cases the $\ldots$ can actually mean remove some terms if $n$ is too small: but the author is assuming that the definition is clear enough that you'll be able to figure out these corner cases.

You might object that this is sloppy notation and that the author should have been more careful or rigorous. On the one hand, if the author's definition is causing confusion, you may be right; on the other, math is all about communication, and sometimes a little bit of informality communicates an idea more clearly than full rigor.

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Normally, $\dots$ is used to omit symbols where there are several related instances of them. It connects some explicitly written-down expressions and trusts that the reader knows how to interpret the omitted expressions from context.

However, this is precisely where it can fail by presenting misleading information or causing ambiguity. A humourous Bell Labs memorandum by E. R. Berlekamp asked

What are the next terms of this sequence? $$2,3,5\dots$$

There are lots of answers just for the fourth term: 7 (primes), 8 (Fibonacci), 10 (sum of all preceding terms), etc. Berlekamp quipped:

It is a remarkable fact that it is often possible to determine a subject's occupation from his answer to a simple question of mathematical induction.

The generic polynomial form presented in this question is no different. The inclusion of an $x^2$ term suggests that linear polynomials do not follow this form even though they do. For this reason, use of $\dots$ is sometimes discouraged in article writing in favour of sigma/pi/set-builder notation, in which the polynomial expression would be written $\sum_{i=0}^na_ix^i$.

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$$P_n=a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x+a_0$$ is the general form.

For $n=0$ we are at the last term that is $$P_0=a_0$$

For $n=1$ we are at the last two terms that is $$P_1= a_1x +a_0$$

For $n=2$ we are at the last three terms that is $$P_2= a_2 x^2+a_1x +a_0$$

And so forth.

No duplication is intended.

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That's not an operator. It's just a placeholder for the other items in the list that follow the same pattern.

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