Despite this is being a very trivial question, I would like to know who was the first (with reference evidence) to show that the following quintic equation has five radical roots: $$x^5 + x + 1 = 0$$

What brings up this question is that "the radical real root for this specific example is impossible to be contained in a much wider range formula given as an independent following question for any polynomial degree as a trinomial polynomial (with all coefficients are zeros or (+/-) one (for only three terms),

however, that question which is an answer itself may not have been received well enough by some mathematicians mentioning that I'm not a professional mathematician which makes hard enough to convey the simplest idea to the professionals in this field

"Is this a NEW CORRECT One real approximated root in a rational form of the following $n$-th degree polynomial equation $$ x^n + x^m = 1,$$ where $(n > m)$ are two distinct positive integers, is given by the following series formula as an approximation where no real root ever exists exactly as we may imagine with our imaginative roots that are strictly associated or derived from that real root that never exists

$$ \begin{align} x &= 1-\frac{1}{n}+\sum_{k=2}^N \frac{(-1)^k \prod_{i=1}^{k-1}(km-in+1)}{k!n^k} \\ &= 1-\frac{1}{n}+\frac{2m-n+1}{2!n^2}- \frac{(3m-2n+1)(3m-n+1)}{3!n^3} +\frac{(4m-3n+1)(4m-2n+1)(4m-n+1)}{4!n^4} -\frac{(5m-n+1)(5m-2n+1)(5m-3n+1)(5m-4n+1)}{5!n^5} +\dotsb \end{align}$$

Where $N$ is a relatively small suitable chosen integer up to our practical needs and capability of making our real roots more sensible (only in rational form) for any other purpose since it is absolutely and perpetually impossible to make the sum of our terms go with no number like infinity since then our real root in our minds would simply become like a ratio of two no existing numbers (each with an uncountable number of digits) which isn't any number, especially that root also is impossible to be constructed exactly by any means

Note that in very rare cases, the above trinomial polynomial has real existing roots in constructible number form and strictly in their surd form (and never in their approximations in decimal or rational forms), when $(n = 2m)$, and $m$ is a power of two



For a reference to the more general solution of this Trinomial equation $$ax^n +bx^m +c = 0$$ click, OK, then click National then type in the search button "solution of equations by power series" at this link http://opac.nl.gov.jo/uhtbin/cgisirsi.exe/zJ7a8B3i0n/MAIN/263930002/123

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    $\begingroup$ I doubt there is a known "first" for the observation $x^5+x+1 = (x^2+x+1)(x^3-x^2+1)$. $\endgroup$ – Gerald Edgar Nov 10 '15 at 14:13
  • $\begingroup$ @GeraldEdgar, and from where did you learn that factorization first (despite the fact that isn't of much significant importance so far? $\endgroup$ – Bassam Karzeddin Jun 18 at 11:41
  • $\begingroup$ I'm still offering the chance s for better answers (if existing ), so would like to wait until the end, where then this modest bounty would be granted to the best available answers as a necessary matter of justice $\endgroup$ – Bassam Karzeddin Jul 11 at 0:46

I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is:

$$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j).$$

It appears to be the first discovered general formula for real roots of the polynomial $x^n + x^m - 1 = 0$. It is not clear whether other radical roots can also be obtained using hypergeometric series.

*I would be interested in a discussion of mathematics forums and how they could be improved, and also how they could coordinate to address different kinds of questions.

I did find a "meta" question here about questions like: "Who first discovered X?". I couldn't participate in that discussion.

The reason for deleting BK's original answer (according to comment) is because it was not on topic of history of mathematics.

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  • $\begingroup$ It is so strange that new discoveries are painful to few mathematicians in power, I appreciate your interest in this discovery, wonder also about the silence of public mathematicians whom must support whatever is true, it could be not visible to them, I wonder!, at any case, It is only a very little result of mine from more general one, also documented in a book (1994), and had its own history from 1986, I can simply fill the word sites of mathematics of this only little formula, and the whole world can't simply refute it, it also contains many secrets I didn't mentioned yet, but any one can $\endgroup$ – Bassam Karzeddin May 11 '16 at 15:26
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    $\begingroup$ This seems to be largely a for-profit forum, although I don't know for sure. I would recommend using Quora for asking questions like this in the future. $\endgroup$ – Gus Wiseman May 19 '16 at 0:41
  • $\begingroup$ But, Quora also deleted the same formula without obvious reasons, it seems to me that it is determined in advance by some authority that they will never consider any thing in mathematics from any public domain unless they are the owners and regardless of its importance", and the purpose is also profit, other wise, a public domain "internet site" can simply produce something that top Journals can't, and this is would be the last nail in their existent purpose, which practically is going to vanish by the time due to huge and very fast world progress in communication technologies $\endgroup$ – Bassam Karzeddin May 19 '16 at 8:17

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