Hi Sarge. Not all polynomials of degree 5 or higher have a full set of solutions, however any polynomial in one variable with constant coefficients will have at least one solution. Unfortunately it has already been proven that there is no general formula or algorithm for finding the exact roots of polynomials of degree 5 or higher. This result is the 'Abel-Ruffini theorem'.
Hi Juan. According to the fundamental theorem of algebra a degree n polynomial could have just 1 solution of multiplicity n, but sure this still counts. You are right that a formula is only ruled out in the case it is based on radicals, but this in turn rules out any algebraic solution. If the recursive approach was followed it seems it would lead to an algebraic solution, no?
Hi Sarge. What you said is true, but it seems unlikely that finding a solution will be as easy as following the recursive method. By following this method I cannot see how limits could be involved, it seems to be heading toward a solution involving the coefficients of the polynomial and taking roots. The Abel-Ruffini theorem does rule out a solution of this form, so perhaps a slightly more complex approach is needed, or just some kind of alteration. Good luck!
Dear Sarge.
This thread has been initiate by me, and in the introductory message I have manifested to every possible contributor the following paragraph.
"The questions written in this thread must be of some significance in order to deserve to be answered in a paper, which can be published in ordinary journals. Indeed, those questions which can be answered in some paragraphs in this web do not fit into this topic. "
If you recognize that you have dispersed one myth, you will agree with me that myths do not fit in this thread. You discussion about Abel-Ruffini is correct, however It should not have occurred if it had made correct use of this thread.
This is why I think that this thread does not benefit anyone. Deleting the thread every myth vanishes.
Best regards,
Dear Hanspeter,
It is the metamorphosis of another that degenerated.
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