RE: factoring quartic over radicals
- To: mathgroup at smc.vnet.net
- Subject: [mg37045] RE: [mg37006] factoring quartic over radicals
- From: "David Park" <djmp at earthlink.net>
- Date: Mon, 7 Oct 2002 05:24:53 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Steve,
You could use the Extension feature of Factor as documented in Help.
expr = x^4 + x^3 + x^2 + x + 1
ans = Factor[expr, Extension -> {1/GoldenRatio}]
(-(1/4))*(-2 - x + Sqrt[5]*x - 2*x^2)*
(2 + x + Sqrt[5]*x + 2*x^2)
You could also use...
Factor[expr, Extension -> {Sqrt[5]}]
It took me some effort to figure out how to manipulate the answer into your
form.
ans /. {x + Sqrt[5]*x -> (2*GoldenRatio)*x,
-x + Sqrt[5]*x -> (2/GoldenRatio)*x}
% /. (-4^(-1))*a_*b_ :> Simplify[-a/2]*Simplify[b/2]
(-(1/4))*(-2 + (2*x)/GoldenRatio - 2*x^2)*
(2 + 2*GoldenRatio*x + 2*x^2)
(1 - x/GoldenRatio + x^2)*(1 + GoldenRatio*x + x^2)
David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/
From: Steve Earth [mailto:SteveE at harker.org]
To: mathgroup at smc.vnet.net
Greetings MathGroup,
My name is Steve Earth, and I am a new subscriber to this list and also a
new user of Mathematica; so please forgive this rather simple question...
I would like to enter the quartic x^4 + x^3 + x^2 + x + 1 into Mathematica
and have it be able to tell me that it factors into
(x^2 + GoldenRatio x + 1) ( x^2 - 1/GoldenRatio x + 1)
What instructions do I need to execute to achieve this output?
-Steve Earth
Harker School
http://www.harker.org/