RE: Exact real solutions of cubic equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg46953] RE: [mg46932] Exact real solutions of cubic equations*From*: "David Park" <djmp at earthlink.net>*Date*: Wed, 17 Mar 2004 02:29:07 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Jonas, First solve a particular case of your equation. sols = With[ {a = -4, b = 3}, Solve[1 + a s + b s^2 + s^3 == 0, s]]; Extract the actual solutions as sols2 and look at their approximate values. sols2 = s /. sols; sols2 // N // Chop {0.692021, -4.04892, 0.356896} The following gets exact values, although maybe they are not in the form you were looking for. If we take approximate values we see that they are real and the same as above. ComplexExpand[Re[#]] & /@ sols2 % // N {-1 + Sqrt[7/3]* Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] + Sqrt[7/3]*Cos[(1/3)*(-Pi + ArcTan[ 1/(3*Sqrt[3])])], -1 - (1/2)*Sqrt[7/3]* Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] - (1/2)*Sqrt[7/3]* Cos[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])] - (1/2)*Sqrt[7]* Sin[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] + (1/2)*Sqrt[7]* Sin[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])], -1 - (1/2)*Sqrt[7/3]* Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] - (1/2)*Sqrt[7/3]* Cos[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])] + (1/2)*Sqrt[7]* Sin[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] - (1/2)*Sqrt[7]* Sin[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])]} {0.692021, -4.04892, 0.356896} David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: JonasB at iui.se [mailto:JonasB at iui.se] To: mathgroup 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