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
>