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

```

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