Re: Exact real solutions of cubic equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg46964] Re: Exact real solutions of cubic equations*From*: drbob at bigfoot.com (Bobby R. Treat)*Date*: Thu, 18 Mar 2004 01:24:49 -0500 (EST)*References*: <c387p7$bij$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In version 5.0.1, at least, this is simple enough: Solve[1 + a s + b s^2 + s^3 == 0 /. {a -> -4, b -> 3}, s]; Simplify@ComplexExpand[s /. %] {-1 + Sqrt[7/3]*Cos[(1/3)*ArcTan[1/(3*Sqrt[3])]] + Sqrt[7]*Sin[(1/3)* ArcTan[1/(3*Sqrt[3])]], -1 - 2*Sqrt[7/3]*Cos[(1/3)*ArcTan[1/(3*Sqrt[3])]], -1 + Sqrt[7/3]*Cos[(1/3)*ArcTan[1/(3*Sqrt[3])]] - Sqrt[7]*Sin[(1/3)*ArcTan[1/(3*Sqrt[3])]]} Bobby JonasB at iui.se wrote in message news:<c387p7$bij$1 at smc.vnet.net>... > Hello, > > I would like to find the _exact_ real roots of some cubic polynomials. > Mathematica seems to have problems determining that a root is real > > Solve[1 + a s + b s^2 + s^3 == 0, s] > > results in three complex solutions for a = -4 and b = 3. FullSimplify does > not help, either it does nothing or it gets stuck, depending on the values > of a and b. I can of course evaluate the solution numerically, but that is > not what I want. Does anyone know of a package that can simplify expressions > with complex numbers? > > Jonas