Re: Exact real solutions of cubic equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg46957] Re: Exact real solutions of cubic equations*From*: "David W. Cantrell" <DWCantrell at sigmaxi.org>*Date*: Wed, 17 Mar 2004 02:29:09 -0500 (EST)*References*: <c387p7$bij$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

JonasB at iui.se wrote: > 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. Granted, they look as if they _might_ not be real. But they are in fact real, and Mathematica can tell you that. Just look at their imaginary parts: In[1]:= s /. Solve[1 - 4*s + 3*s^2 + s^3 == 0, s]; Table[FullSimplify[Im[%[[i]]]], {i, 1, 3}] Out[1]= {0, 0, 0} > FullSimplify > does not help, either it does nothing or it gets stuck, depending on the > values of a and b. I think you're wanting something which is theoretically impossible, namely, to have the three real solutions expressed in terms of radicals using just real numbers. It's been known for centuries that this cannot be done. This situation is the _casus irreducibilis_. (For more information, do a Google search for "casus irreducibilis".) OTOH, the three real solutions can be expressed using just real numbers if trig functions and their inverses are allowed. As an example, for 1 - 4*s + 3*s^2 + s^3 == 0, the three real solutions are given in the following table: In[2]:= Table[-1 + (-1)^(n + 1)*2*Sqrt[7/3]*Cos[(1/3)*(ArcCot[3*Sqrt[3]] + n*Pi)], {n, 0, 2}] Out[2]= {-1 - 2*Sqrt[7/3]*Cos[(1/3)*ArcCot[3*Sqrt[3]]], -1 + 2*Sqrt[7/3]*Cos[(1/3)*(Pi + ArcCot[3*Sqrt[3]])], -1 - 2*Sqrt[7/3]*Cos[(1/3)*(2*Pi + ArcCot[3*Sqrt[3]])]} In[3]:= N[%] Out[3]= {-4.048917339522305, 0.35689586789220984, 0.6920214716300952} HTH, David Cantrell