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

*To*: mathgroup at smc.vnet.net*Subject*: [mg54643] Re: [mg54630] finding roots of 1 + 6*x - 8*x^3*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Fri, 25 Feb 2005 01:18:41 -0500 (EST)*Reply-to*: hanlonr at cox.net*Sender*: owner-wri-mathgroup at wolfram.com

The result that you obtained is real and in squart root form Solve[1 + 6*x - 8*x^3==0,x]//Simplify {{x -> (2*2^(1/3) + (2 + 2*I*Sqrt[3])^(2/3))/(4*(1 + I*Sqrt[3])^(1/3))}, {x -> (-(1/2))*((1/2)*(1 + I*Sqrt[3]))^(2/3) + (1/4)*I*((1/2)*(1 + I*Sqrt[3]))^(1/3)* (I + Sqrt[3])}, {x -> (-2 + 2*I*Sqrt[3] - 2^(1/3)*(1 + I*Sqrt[3])^(5/3))/ (4*2^(2/3)*(1 + I*Sqrt[3])^(1/3))}} %//N//Chop {{x -> 0.9396926207859084}, {x -> -0.766044443118978}, {x -> -0.17364817766693055}} Reduce[1 + 6*x - 8*x^3==0,x]//N x == -0.766044443118978 || x == -0.17364817766693036 || x == 0.9396926207859084 Bob Hanlon > > From: "Kennedy" <kennedy at oldnews.org> To: mathgroup at smc.vnet.net > Date: 2005/02/24 Thu AM 03:21:35 EST > To: mathgroup at smc.vnet.net > Subject: [mg54643] [mg54630] finding roots of 1 + 6*x - 8*x^3 > > 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 > >

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

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