[newbie question] solving fifth order polynomial equation
Subject: [newbie question] solving fifth order polynomial equation
From: sen1 at math.msu.edu
Date: Wed, 17 Jan 2007 11:06:32 -0500 (EST)
It seems that "realroots" with no other argument gives rational
solutions.
(%i19) realroots(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5);
44559791 13723459 11184811
(%o19) [s = - --------, s = - 1, s = - --------, s = - --------,
33554432 33554432 33554432
8825613
s = -
--------]
33554432
I presume, but did not check, that this means that the roots are
rational.
Or, are these just approximate values?
-sen
On Wed, 17 Jan 2007, Stavros Macrakis wrote:
> On 1/17/07, Hugo Coolens <coolens at kahosl.be> wrote:
>> When solving the following equation
>> solve(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5=0);
>>
>> Maxima shows two real solutions and three complex ones in stead of five
>> real solutions.
>
> Actually, Maxima is showing five real solutions -- it's just that the
> *form* of some of the real solutions includes %i. To simplify these
> expressions to purely real expressions, use radcan:
>
> radcan(solve(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5=0)) =>
> [s = -1/3,s = -1,
> s = -(sqrt(3)*sin((atan(9/(13*sqrt(3)))-%pi)/3)
> +cos((atan(9/(13*sqrt(3)))-%pi)/3)+2)
> /3,
> s = (sqrt(3)*sin((atan(9/(13*sqrt(3)))-%pi)/3)
> -cos((atan(9/(13*sqrt(3)))-%pi)/3)-2)
> /3,s = (2*cos((atan(9/(13*sqrt(3)))-%pi)/3)-2)/3]
>
> If all you care about is the numeric values of the real solutions, you
> can use realroots to get approximations to any desired precision, even
> for non-factorizable polynomials:
>
> realroots(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5=0, 10^-30)
>
> realroots(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5=0, 10^-30);
> [s = -3366842668502188682920157364511
> /2535301200456458802993406410752,s = -1,
> s = -1036915250298305044736277999857
> /2535301200456458802993406410752,
> s = -845100400152152934331135470251
> /2535301200456458802993406410752,
> s = -666844482112423878330377457135
> /2535301200456458802993406410752]
> bfloat(%),fpprec:30;
> [s = -1.327985277605681767796032025b0,s = -1.0b0,
> s = -4.08990951493896474542054282001b-1,
> s = -3.33333333333333333333333333333b-1,
> s = -2.63023770900421757661913692994b-1]
>
> That one was factorizable into polynomials of degree <= 4, so roots
> can be given as radical expressions:
>
> factor(1+11*s+45*s^2+84*s^3+70*s^4+21*s^5) =>
> (s+1)*(3*s+1)*(7*s^3+14*s^2+7*s+1)
>
> Example not factorizable over the rationals:
> realroots(x^5+x+3) => [x = -38017089/33554432]
>
> -s
> _______________________________________________
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> Maxima at math.utexas.edu
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>
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