Re: Factoring question

*To*: mathgroup at smc.vnet.net*Subject*: [mg35482] Re: Factoring question*From*: Selwyn Hollis <slhollis at earthlink.net>*Date*: Tue, 16 Jul 2002 04:49:39 -0400 (EDT)*References*: <agbfhl$je9$1@smc.vnet.net> <200207130748.DAA08575@smc.vnet.net> <agrjfj$g9n$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Here's how this problem is suppose to play out: You want to factor x^7 + x^5 + x^4 + -x^3 + x + 1. The rational root theorem tells you that the only possible rational roots are -1 and 1. You check those and find that -1 works. Therefore x+1 is a factor. Now you divide to find the other factor, which turns out to be 1-x^3+2x^4-x^5+x^6. Since neither -1 nor 1 is a root of this factor, you're DONE---in the sense that you can't factor any more over the rational numbers. To go further, use NSolve to find the other six roots. None of them are real; so the previous factorization is complete over the irrationals as well. Selwyn Hollis slhollis at mac.com http://www.math.armstrong.edu/faculty/hollis Ken Levasseur wrote: > Steve: > > I assume that the problem was to solve x^7 + x^5 + x^4 + -x^3 + x + 1=0. If > so, one of the basic factoring theorems is that if a polynomial over the > integers like this one has a rational root r/s, then r must divide the > constant term and s must divide the leading coefficient. So in this problem, > +/-1 are the only possible rational roots and so the (x+1) factor would be > found this way. I'm sure that Mathematica checks this almost immediately. > As for the remaining 6th degree factor, I'm not certain how Mathematica > proceeds, but if you plot it, it clearly has no linear factors. > > Ken Levasseur > > > Steven Hodgen wrote: > > >>"DrBob" <majort at cox-internet.com> wrote in message >>news:agbfhl$je9$1 at smc.vnet.net... >> >>>Factor[x^7 + x^5 + x^4 + -x^3 + x + 1] // Trace >> >>This doesn't do it. It only traces the initial evaluation, and then simply >>displays the factored result with no intermediate factoring steps. >> >>Thanks for the suggestion though. >> >> >>>Bobby >>> >>>-----Original Message----- >>>From: Steven Hodgen [mailto:shodgen at mindspring.com] To: mathgroup at smc.vnet.net >> >> >>>Subject: [mg35482] Factoring question >>> >>>Hello, >>> >>>I just purchased Mathematica 4.1. I'm taking precalculus and wanted to >> >>try >> >>>a tough factoring problem, since the teacher couldn't do it either. >>>Mathematica get's the correct answer, but I'm interrested in seeing how it >>>got there. Is there a way to turn on some sort of trace feature where it >>>shows each step it used to get the the final result? >>> >>>Thanks! >>> >>>--Steven >>> >>> >>> >>> >>> >>> >>> >> > >

**References**:**Re: Factoring question***From:*"Steven Hodgen" <shodgen@mindspring.com>