Fwd: factor n^4 + 4 - guessing

[Andres Santana]

Forwarding to the group to share... Read bellow.

---------- Forwarded message ----------
From: Andres Santana <andres....@gmail.com>
Date: Sun, Nov 21, 2010 at 9:28 AM
Subject: factor n^4 + 4 - guessing
To: Carlos Encarnacion <encarnaci...@gmail.com>


I first took this as example:

n^4 - 4 = (n^2 + 2) (n^2 - 2)

Then I tried to do the same

assuming n^4 + 4 = (n^2 + 2) (n^2 + 2) but, expanding the factorization I get that:

(n^2 + 2) (n^2 + 2) = (n^2 +2)^2 = n^4 + 4n^2 + 4

Here n^4 + 4n^2 + 4 I see that I have one term that I don't want 4n^2 and I know this 4n^2 = (2n)(2n) so, since I want eliminate that term I added this in both with inverted signs

(n^2 + 2 + 2n) (n^2 + 2 - 2n) after ordering the term I get (n^2 + 2n + 2) (n^2 - 2n + 2)

Then again I expanded this and I get

n^4 - 2n^3 + 2n^2 + 2n^3 - 4n^2 + 4n + 2n^2 -4n + 4

reducing I get

n^4 + 4

So, n^4 + 4 = (n^2 + 2n + 2) (n^2 - 2n + 2)

I did guessed but It worked out. Your comments are expected.

--
arsh

--
arsh