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Posting to Mma mailing list
*To*: mathgroup at yoda.physics.unc.edu
*Subject*: Posting to Mma mailing list
*From*: cowen at acsu.buffalo.edu (Michael Cowen)
*Date*: Fri, 22 May 92 11:16:06 EDT
I want to post the following message to the Mathematica mailing list, but
I do not know how to do it (at my site we can only post locally to this list):
Subject: Factoring x^4 + n (Beginner's Guide to Mma v.2, Ch. 12)
Chapter 12 of Beginner's Guide to Mathematica version 2.0 by Gray and Glynn
"Can I explore mathematics with Mathematica" has the following statement:
>We asked a number of people if x^4 + 64 can be factored without complex
>numbers. Most said no. Let's see what Mathematica says:
> Factor[x^4 + 64]
> (8 - 4x + x^2)(8 + 4x + x^2)
>A big surprise for many of us!
Several pages of computations then follow illustrating when x^4 + n can
be factored, for some special cases of n . This example seems to miss
several points. First of all, any real fourth degree polynomial factors
into a product of two real quadratics, so the question has nothing to do
with complex numbers. What Gray and Glynn should have asked was can this
be factored over the integers?
Now
(x^2 + a^2 + b x)(x^2 + a^2 - b x) = (x^2 + a^2)^2 - b^2 x^2 =
x^4 + a^4 + (2 a^2 - b^2)x^2
So this equals x^4 + a^4 if and only if b^2 = 2 a^2 , that is
x^4 + n factors over the integers if and only if n = b^4 /4 for b an
integer. So b is necessarily even, b = 2m, and hence n = 4 m^4 for m
any integer. Indeed,
x^4 + 4 m^4 = (x^2 + 2 m^2 + 2 m)(x^2 + 2 m^2 - 2 m) .
Surely this can be illustrated using Mathematica in a more focused way than
is done in Beginner's Guide.
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