# Finding real roots for an eigth-degree polynomial

The context for this problem is identical to my first question. Unfortunately my previous result for $L$ was inexact since it was a relatively simple approximation. This time I have managed to produce a very convincing approximation for $L$. Alike last time, solving for $W$ provides the $n$th solution if you find the $n$th root of a specific [octive] polynomial:

$96x^8-192Lx^7+(768π^2 A^2+128L^2 ) x^6+(-64L^3-640A^2 π^2 L) x^5+(2655π^4 A^4-640L^2 π^2 A^2+32L^4 ) x^4+(384A^2 L^3 π^2-578A^4 Lπ^4 ) x^3+(4476π^6 A^6-2760L^2 π^4 A^4+128L^4 π^2 A^2 ) x^2+(456A^6 Lπ^6-278A^4 L^3 π^4 )x+(3136π^8 A^8-3700L^2 π^6 A^6+1089L^4 π^4 A^4 )=0$

Since my investigation is based on the assumption that there is only one value for $W$, given the values of $A$ and $L$, ergo there should only be two real roots (one should be the positive value for $W$ and the other should be the negative value for $W$.)

My question is, can you find the positive real root for the eight order polynomial above?

• Do you think it has only two Real roots for any {L, A}? – Dr. belisarius Sep 5 '14 at 21:51
• Yes, [I'm not sure if this matters but] provided {L,A}>0 – J.D'Almbert Sep 5 '14 at 21:53
• Might want to provide actual code. – Daniel Lichtblau Sep 5 '14 at 22:07
• Solve[l= (√(4 π^2 A^2+W^2 )+W)/2 ((3(√(4 π^2 A^2+W^2 )-W)^2)/(√(4 π^2 A^2+W^2 )+W)(10√(4 π^2 A^2+W^2 )+10 W+√(√(4 π^2 A^2+W^2 )^2 W^2+14 √(4 π^2 A^2+W^2 ) W)) +1), W] – J.D'Almbert Sep 5 '14 at 22:13
• 96 x^8 - 192 L x^7 + (768 [Pi]^2 A^2 + 128 L^2) x^6 + (-64 L^3 - 640 A^2 [Pi]^2 L) x^5 + (2655 [Pi]^4 A^4 - 640 L^2 [Pi]^2 A^2 + 32 L^4) x^4 + (384 A^2 L^3 [Pi]^2 - 578 A^4 L [Pi]^4) x^3 + (4476 [Pi]^6 A^6 - 2760 L^2 [Pi]^4 A^4 + 128 L^4 [Pi]^2 A^2) x^2 + (456 A^6 L [Pi]^6 - 278 A^4 L^3 [Pi]^4) x + (3136 [Pi]^8 A^8 - 3700 L^2 [Pi]^6 A^6 + 1089 L^4 [Pi]^4 A^4) – J.D'Almbert Sep 5 '14 at 22:23

Let's define a polynomial p of one variable x depending on two parameters A and L:

p[x_, A_, L_] :=
96 x^8 − 192 L x^7 + (768 π^2 A^2 + 128 L^2) x^6
+ (−64 L^3 − 640 A^2 π^2 L) x^5 + (2655 π^4 A^4 − 640 L^2 π^2 A^2 + 32 L^4) x^4
+ (384 A^2 L^3 π^2 − 578 A^4 L π^4) x^3
+ (4476 π^6 A^6 − 2760 L^2 π^4 A^4 + 128 L^4 π^2 A^2) x^2
+ (456 A^6 L π^6 − 278 A^4 L^3 π^4) x
+ (3136 π^8 A^8 − 3700 L^2 π^6 A^6 + 1089 L^4  π^4 A^4)


An appropriate tool for finding positive solutions is Solve (one can use Reduce also) with the MaxExtraConditions -> All option, moreover we need the assumption x > 0 and aditionally domain specification Reals, otherwise A and L might be complex in general, making the task too involved.
So we have:

TraditionalForm[ sol = Solve[ p[x, A, L] == 0 && x > 0, x, Reals,
MaxExtraConditions -> All]] There are three positive real solutions under quite involved conditions ( the second arguments of ConditionalExpression), however the first solution is in a zero measure subset of variable space A , L thus in fact generically there are only two positive solutions.
We can verify that the conditions are not satisfied e.g. for A == 1 && L ==1, e.g.

Simplify[ sol, A == 1 && L == 1]

 {{x -> Undefined}, {x -> Undefined}, {x -> Undefined}}

• Does this just proves that there are two positive real roots? How can I find these roots as a f[A,L]? – J.D'Almbert Sep 5 '14 at 22:51
• @J.D'Almbert I updated my answer. Root's are symbolic representations of exact solutions, under specific conditions one might apply ToRadicals to Root objects finding solutions in terms of radicals but in general you cannot expect such solutions because it is mathematically impossible to find solutions in terms of radicals for polynomials of order higher than $4$. – Artes Sep 5 '14 at 23:04
• It is important to add that L>>A, e.g. when W=3 and A=2, L=8.7532759347618317962647585592473591641754313583030763 – J.D'Almbert Sep 6 '14 at 9:03
• What is $W$?. Edit your question and make it clear since I'm not sure what you really want if my answer does not satisfy you needs. – Artes Sep 6 '14 at 9:10
• When you simplify "sol" could you make A=2 and L=8.7532759347618317962647585592473591641754313583030763 – J.D'Almbert Sep 6 '14 at 9:16