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RE: factoring quartic over radicals


It's a little ugly, but here's my solution:

x^4 + x^3 + x^2 + x + 1
Simplify@Factor[%, Extension -> {GoldenRatio, 1/GoldenRatio}]
Collect[%[[2]]/3, x]Collect[%[[3]]/3, x]
% /. Sqrt[5] -> 2GoldenRatio - 1
% // Simplify
Collect[#, x] & /@ %
% /. {1 - GoldenRatio -> -1/GoldenRatio}
FullSimplify@Expand@%

(The last line is a check.)

Bobby

-----Original Message-----
From: Steve Earth [mailto:SteveE at harker.org] 
To: mathgroup at smc.vnet.net
Subject: [mg37048] [mg37006] factoring quartic over radicals

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/





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