Re: finding roots of 1 + 6*x - 8*x^3

*To*: mathgroup at smc.vnet.net*Subject*: [mg54663] Re: [mg54630] finding roots of 1 + 6*x - 8*x^3*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Fri, 25 Feb 2005 01:19:22 -0500 (EST)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*References*: <200502240821.DAA13324@smc.vnet.net>*Reply-to*: murray at math.umass.edu*Sender*: owner-wri-mathgroup at wolfram.com

A similar question, about a quadratic, just appeared in this list, with quite a few variants of essentially the same response -- use ComplexExpand. One of the simplest, for your example, is: ComplexExpand[x /. Solve[1 + 6x - 8x^3 == 0, x]] Kennedy wrote: > Hello All, > > I am trying to find the roots of > 1 + 6*x - 8*x^3. > > Roots[1+6*x-8*x^3==0,x] yields this ugly thing: > (made uglier by my converting to InputForm) > > x == ((1 + I*Sqrt[3])/2)^(1/3)/2 + > 1/(2^(2/3)*(1 + I*Sqrt[3])^(1/3)) || > x == -((1 - I*Sqrt[3])*((1 + I*Sqrt[3])/2)^(1/3))/4 - > ((1 + I*Sqrt[3])/2)^(2/3)/2 || > x == -(1 - I*Sqrt[3])/(2*2^(2/3)*(1 + I*Sqrt[3])^ > (1/3)) - (1 + I*Sqrt[3])^(4/3)/(4*2^(1/3)) > > This monstrosity is chock full of imaginaries, > even though I know all three roots are real. > > I tried Solve too but got the same thing, except > in the form of a set of replacements. My guess is > that Solve just calls Roots when handed a poly- > nomial. > > When I ran the above through FullSimplify, I > got three "Root" objects, the upshot of which is > that the roots of the polynomial are indeed the > Roots of said polynomial. Huh. > > What command can I use to get the roots into > a form that are > (a) purely real, and > (b) in radical form? > > Thanks, > Kennedy > > PS. Mathematica 4.2 > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**finding roots of 1 + 6*x - 8*x^3***From:*"Kennedy" <kennedy@oldnews.org>