## How approximate a sextic polynomial to a lower degree polynomial

Hi all,

I have been stopped by a sextic (6th degree) polynomial in my research. I need to find the biggest positive root for this polynomial symbolically, and since its impassible in general, I came up with this idea, maybe there is a way to approximate this polynomial by a lower degree polynomial which is solvable.

κ2/112 (A2 ) u62/16 (A2 ) u52/20 (1/2 B2+3 A2 ) u42/8 (A2+B2 ) u3-((ω2-B2 κ2)/6) u22 κ2 ω2=0

this polynomial is come from a nonlinear PDE related to waves.
κ, A, B, v, ω , u are not constant.

I appreciate any helpful comment or solution.
Thanks,
Romik
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 Mentor Divided by u6, this is a polynomial of 6th order in (1/u), where you look for the smallest positive root. Depending on the parameters, a taylor expansion or something similar might give some reasonable analytic approximation.
 Thanks for the reply, biggest or smallest positive root, that's not the main issue here, I need to find an approximate root based on variables, reduce from 6th degree to lets say 4th degree which I can solve it exactly.

Mentor

## How approximate a sextic polynomial to a lower degree polynomial

Well, the biggest root in your original equation would be the smallest root in my modified equation.
On second thought, my idea with a taylor approximation around the origin would simply neglect the absolute term. The remaining polynomial can be expressed as u^2 P(u) where P has order 4, so there are analytic solutions. It might be interesting to improve this approximation with one or two steps of Newton afterwards ;).
 I don't know if this would help, but... you could try the substitutions\begin{align*} x &= A^2 u^2 \\ y &= B^2 u^2 \\ z &= \omega^2 v^2 \\ s &= \omega^2 u^2 \end{align*}to obtain the possibly simpler equation$$\frac {\kappa^2 x} {112} u^4 + \frac {\kappa^2 x} {16} u^3 + \left( \frac {\kappa^2 (6x+y)} {40} \right)u^2 + \frac{\kappa^2 (x+y)} 8 u - \frac{s-\kappa^2 y} 6 + \kappa^2 z = 0$$ If you somehow manage to obtain values for $u,x,y,z,s$, then $\omega = \pm\sqrt{\displaystyle\frac s {u^2}}$, and the values for $A,B,v$ can be solved for similarly. (I was trying to put also $\kappa$ into the substitutions for $x,y,z$, but then I can't find the original variables back. Unless you have an extra constraint on them.)
 thanks mfb for you helpful comments. can you explain more about Newton method, how could I apply it on my equation? I use Mathematica! with Series function, I am able to truncate my original polynomial to 4th degree, now how should I apply Newton since I don't have numerical root and my roots are symbolical? thank you Dodo for your reply, did you know you put your 666th post on this thread? So good luck to me :D

Mentor
 I am able to truncate my original polynomial to 4th degree
Great, that has analytic solutions, and you don't need Newton.

 now how should I apply Newton since I don't have numerical root and my roots are symbolical?
Let x be the approximate position of the root, f(x) be the function value there and f'(x) its derivative. Both f(x) and f'(x) are easy to express symbolically. A (hopefully) better approximation for the root is then given by x-f(x)/f'(x).

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