4th degree equations

Hi, can anyone help me with this problem? We need the smallest positive real root of this equation

a*x^4+b*x^3+c*x^2+d*x+e=0,

where a>0, b<0, c>0, d<0 and e>0. As Descartes said, in that case this equation has at least 2 positive real roots.

Thank you for your attention.

Answer 1

[Walter Roberson]

r = roots([a,b,c,d,e]);
r(imag(r)~=0) = [];  %remove complex
r(r <= 0) = [];  %remove non-positive
min(r)  %now it is the smallest positive real root

[Asatur Khurshudyan]

Thank you very much Mr. Robertson.

Could you, please, find the root in the case of a>0, b<0, c>0, d<0 and e>0.

[Walter Roberson]

Are the coefficients symbolic variables or actual values? The above code is for actual values. If you are asking for the general symbolic solution, then sorry, I do not know if it is possible; I can see from unconstrained solution to the roots that it would be quite quite messy if it can be done.

[Asatur Khurshudyan]

Yes, unfortunately they are symbolic variables, otherwise, there is Ferrary`s method for its solving, but it`s very difficult. If you have a bit of free time, please try it for us for symbolic coefficients, we shall be very grateful.

[Walter Roberson]

I think I do not know enough polynomial theory to come up with a useful answer on this, sorry.

[Asatur Khurshudyan]

No, I mean with Matlab. If there exist exact method, than it must be done by mathlab, I suppose. All right, as you wish.