Given the following polynomial:
$ p(x)=25x^8 +85x^7 +181x^6 +238x^5 +226x^4 +142x^3 +61x^2 +13x+1 $
Which factors as $(x^2 + x + 1)^3(5x + 1)^2$, (It has a triple complex root and their conjugates and a double real root) I am trying to obtain the roots of the perturbed polynomial
$h(x) = p(x) + 10^{−8}x^5$
I know obtaining roots of a polinomial is an ill-posed problem. I am using Maehly's method in Matlab.
Matlab gives the roots:
-5.000568859054557e-01 + 8.664727042796275e-01i
-5.000568859054557e-01 - 8.664727042796275e-01i
-5.003588172119005e-01 + 8.657523017471233e-01i
-5.003588172119005e-01 - 8.657523017471233e-01i
-4.995842968797133e-01 + 8.658512052028565e-01i
-4.995842968797133e-01 - 8.658512052028565e-01i
-2.000004647944056e-01 + 0.000000000000000e+00i
-1.999995352114569e-01 + 0.000000000000000e+00i
Using Maehly's method I obtain poor convergence. How can I deduce if the roots are multiple or they are very close?
I have already evaluated the roots in both the derivative polynomial $Dp$ and $p$ and, my conclusions are that none of the roots is double, because I obtain, for example:
$$ p(-2.000000000026995e-01)=-3.200106846179551e-12$$ and $$ Dp(-2.000000000026995e-01)=1.776356839400251e-15$$
Do you think it is safe to asume that this root is not multiple? If so, How should I compute the root that is very close to this one?