August 24th, 2009, 12:27 AM  #1 
Newbie Joined: Aug 2009 Posts: 1 Thanks: 0  Factor x^4 + 1
Hi, How can I factor x^4 + 1 using two polynomial of grade 2 with real coefficients? I managed to do with complex numbers but I can't find noncomplex polynomials is there a method? Thanks 
August 24th, 2009, 05:39 AM  #2 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Factor x^4 + 1
Here's how I would do it: x^4 + 1 = (x^2 + Bx + C)(x^2 + Dx + E) = x^4 + (B + D)x^3 + (BD + C + E)x^2 + (BE + CD)x + CE Equating by power: B + D = 0 BD + C + E = 0 BE + CD = 0 CE = 1 So B = D BE + CD = 0 > DE + CD = 0 > D(C  E) = 0 > C = E Since CE = 1 and C = E, either C = 1 or C = 1. Assume C = 1. Then BD + C + E = 0 > BD = 2 > B = sqrt(2). This gives: x^4 + 1 = (x^2 + sqrt(2) x + 1)(x^2  sqrt(2) x + 1) which is the factorization you want. 
August 25th, 2009, 08:19 AM  #3 
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Factor x^4 + 1
Easier to do as a difference of squares, but a rose is a rose is a rose ... if there are complex numbers then there are complex numbers ..period.

August 25th, 2009, 09:20 AM  #4  
Global Moderator Joined: Dec 2006 Posts: 20,828 Thanks: 2160  Quote:
For example, (x + (1 + i)/?2)(x + (1 i)/?2) = x² + ?2x + 1.  

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