       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:=
SolveAlways[24-50*x+35*x^2-10*x^3+x^4==(x^2+b*x+c)(x^2+e*x+f),x]
Out=
{{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

In:=
SolveAlways[x^4+x^2+1==(x^2+b*x+c)(x^2+e*x+f),x]
Out=
{{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, c -> -1, e -> I Sqrt, f -> -1},

{b -> I Sqrt, c -> -1, e -> -I Sqrt, f -> -1}}

If you want only real answers you can do (for example):

In:=
Select[%, ({b, c, e, f} /. #1) \[Element] Reals & ]
Out=
{{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 ?