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MathGroup Archive 2002

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RE: factoring quartic over radicals

  • To: mathgroup at
  • Subject: [mg37045] RE: [mg37006] factoring quartic over radicals
  • From: "David Park" <djmp at>
  • Date: Mon, 7 Oct 2002 05:24:53 -0400 (EDT)
  • Sender: owner-wri-mathgroup at


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

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

From: Steve Earth [mailto:SteveE at]
To: mathgroup at

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

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