My Math Forum Factor x^4 + 1

 Algebra Pre-Algebra and Basic Algebra Math Forum

 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 non-complex 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: 19,704
Thanks: 1804

Quote:
 Originally Posted by profetas I managed to do with complex numbers but I can't find non-complex polynomials . . .
If complex numbers are used at all, you should know that the complex linear factors come in conjugate pairs whose products are the real quadratic factors.
For example, (x + (1 + i)/?2)(x + (1- i)/?2) = x² + ?2x + 1.

 Tags factor

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Albert.Teng Algebra 7 June 16th, 2012 04:30 AM daigo Algebra 3 June 14th, 2012 05:38 PM HellBunny Algebra 3 February 18th, 2012 10:31 AM Eminem_Recovery Algebra 11 June 19th, 2011 08:50 PM haebin Calculus 2 September 14th, 2009 09:25 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top