4

Find all polynomials $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ such as:

1) $(a_0,a_1,a_2,...,a_n)$ is a permutation of $(0,1,2,...,n)$

2) $P(x)$ has $n$ rational roots.

Road Human
  • 1,012
  • 2
    Why? Please give some indication of where this question comes from, why it's worth studying, what progress you have made on it, what level of answer you are expecting, etc. – Gerry Myerson Jan 06 '15 at 15:34
  • @GerryMyerson None of that is necessary. This problem is beautiful and that is reason enough. – Pp.. Jan 06 '15 at 15:36
  • 2
    @Pp.., that's reason enough to work on it. It's not reason enough to post an answer. It also doesn't tell us what kind of answer will be appreciated by OP. – Gerry Myerson Jan 06 '15 at 15:39
  • @Pp.. You don't have an answer. Don't diverge from the math. – Pp.. Jan 06 '15 at 15:40
  • @RoadHuman So far I have only info about $a_0$. Notice it doesn't have positive roots. If we put $-x$ instead of $x$ we see that there are less than $n-1$ sign changes (since one of the coefficients is zero). By Descartes's rule of signs it must have less than $n$ positive roots (the transformed polynomial). Therefore we need zero to be a root, i.e. $a_0=0$. – Pp.. Jan 06 '15 at 15:55
  • @GerryMyerson See? It was reason enough to post an answer. Notice also how there there were people like you asking annoying irrelevant questions to the OP (with their groupies), none of them solved the problem. – Pp.. Jan 06 '15 at 16:07
  • 1
    @Pp.., you've been here for 8 days. There are reasons, good reasons, why people here ask for more information when questions are posted with no source, no hint of any effort on the part of the person asking, and no indication of the level of answer expected. Maybe when you've been here a bit longer, you'll have a better understanding of the reasons for this. – Gerry Myerson Jan 06 '15 at 21:42
  • @GerryMyerson You are using faulty assumptions to draw your conclusions. The relation (human,account) is neither a function nor it has to be injective. I have used this website since its creation and understand perfectly well why people do what you do. I wouldn't call it good reasons. – Pp.. Jan 06 '15 at 23:32
  • 1
    @Pp.. I'm sorry to see that you are so ashamed of your earlier work on this site that you need to disassociate yourself from it by changing your screen name, and I fear it doesn't bode well for your future participation that you choose to hide your identity behind your Pp. – Gerry Myerson Jan 07 '15 at 16:43
  • @GerryMyerson The protection of my identity sets me free to call upon those that do wrong, and makes me proud that the help I give I give for nothing in return. My two shames are once begin rough to a student having a hard time to understand, and having colleagues that give a bad name to my profession in this site without being ashamed. – Pp.. Jan 07 '15 at 17:12

0 Answers0