Approximating solutions to any(?) perturbed polynomial of any degree by limiting ratios.

(*Mathematica*)(*start*)Clear[A, B, a, b, x];
a = {1, 1, 1, 1, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0};
b = (Range[Length[a]] - 1)!;
a = a*b;
nn = Length[a];
d = Max[Flatten[Position[Sign[Abs[a]], 1]]]
A = Table[
   Table[If[n >= k, Binomial[n - 1, k - 1]*a[[n - k + 1]], 0], {k, 1,
     Length[a]}], {n, 1, Length[a]}];
b = Inverse[A][[All, 1]];
x = N[Table[(n - 1)*b[[n - 1]]/b[[n]], {n, nn - 8, nn - 1}], 30]
perturbation = -1;
x - perturbation
Sum[a[[k + 1]]/k!*x^k, {k, 0, d}]
Clear[x];
polynomial = Sum[a[[k + 1]]/k!*x^k, {k, 0, d}]
a1 = CoefficientList[
   Expand[Sum[a[[k + 1]]/k!*(x + perturbation)^k, {k, 0, d}]],
   x]*(Range[d] - 1)!
Clear[A, B, a, b, x];
a = Flatten[{a1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
     0, 0, 0, 0, 0}];
nn = Length[a];
d = Max[Flatten[Position[Sign[Abs[a]], 1]]]
A = Table[
   Table[If[n >= k, Binomial[n - 1, k - 1]*a[[n - k + 1]], 0], {k, 1,
     Length[a]}], {n, 1, Length[a]}];
b = Inverse[A][[All, 1]];
x = N[Table[(n - 1)*b[[n - 1]]/b[[n]], {n, nn - 8, nn - 1}], 30]
"perturbed x polynomial"
Sum[a[[k + 1]]/k!*x^k, {k, 0, d}]
x = x + perturbation
"unperturbed x polynomial"
polynomial
Clear[x];
Sum[a[[k + 1]]/k!*x^k, {k, 0, d}]
(*end*)