Re: Re: Fourth degree polynomial
- To: mathgroup at smc.vnet.net
- Subject: [mg29310] Re: [mg29285] Re: [mg29251] Fourth degree polynomial
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Wed, 13 Jun 2001 03:10:44 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
There is an easy way to do this which does not require finding the roots of the polynomial explicitely (in fact there are several such ways but I give only the simplest one). In[7]:= SolveAlways[24-50*x+35*x^2-10*x^3+x^4==(x^2+b*x+c)(x^2+e*x+f),x] Out[7]= {{b -> -7, c -> 12, e -> -3, f -> 2}, {b -> -6, c -> 8, e -> -4, f -> 3}, {b -> -5, c -> 4, e -> -5, f -> 6}, {b -> -5, c -> 6, e -> -5, f -> 4}, {b -> -4, c -> 3, e -> -6, f -> 8}, {b -> -3, c -> 2, e -> -7, f -> 12}} Although it looks like there are 6 solutions of course there are only three, since the factors simply got switched. The same method works of course in the other examples mentioned in this thread, e.g. In[8]:= SolveAlways[x^4+x^2+1==(x^2+b*x+c)(x^2+e*x+f),x] Out[8]= {{b -> -1, c -> 1, e -> 1, f -> 1}, 1/3 2/3 {b -> 0, c -> (-1) , e -> 0, f -> -(-1) }, 2/3 1/3 {b -> 0, c -> -(-1) , e -> 0, f -> (-1) }, {b -> 1, c -> 1, e -> -1, f -> 1}, {b -> -I Sqrt[3], c -> -1, e -> I Sqrt[3], f -> -1}, {b -> I Sqrt[3], c -> -1, e -> -I Sqrt[3], f -> -1}} If you want only real answers you can do (for example): In[9]:= Select[%, ({b, c, e, f} /. #1) \[Element] Reals & ] Out[9]= {{b -> -1, c -> 1, e -> 1, f -> 1}, {b -> 1, c -> 1, e -> -1, f -> 1}} -- Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/ http://sigma.tuins.ac.jp/~andrzej/ > on 8/06/01 10:15, Stephane Redon at Stephane.Redon at inria.fr wrote: > > Hello everybody, > > I've got a fourth degree polynomial which I would like to break into two > second order polynomials. Unfortunately, the Factor function doesn't manage > to do it, probably because it attempts to find all the roots of my > polynomial. Is there a way to do this WITHOUT finding the roots ? > Thanks in advance > > Stephane >