Extract different part of a polynomial with square root $a(x)+b(x)\sqrt{c(x)}$

Question

Say I have an expression like $a(x)+b(x)\sqrt{c(x)}+d(x)\sqrt{e(x)}+f(x)g(\sin(x))$, where a(x) to f(x) mean polynomials of $x, g(x)$ is a polynomial of $sin(x)$. I wonder how to extract the different terms, e.g., $b(x), c(x), a(x), f(x)$?

If the above expression is too complex, how about the simplified version $a(x)+b(x)\sqrt{c(x)}$? This is what I am primarily concerned about.

The reason I'm concerned with this is that I'd like to get rid of the square root to get a polynomial, to solve the equation. I tried using the eliminate command as suggested by example using Eliminate, but somehow it returns very large coefficients like around $10^{260}$ for all the coefficients.

-1.52537-2.54462 Cos[\[Delta]]^3+Cos[\[Delta]]^2 (2.94197 -1.35683 Sin[\[Delta]])+2.44869 Sin[\[Delta]]-0.853005 Sin[\[Delta]]^2+Cos[\[Delta]] (4.69792 -2.41511 Sin[\[Delta]]-0.180872 Sin[\[Delta]]^2)+Sqrt[1/25-(0.102901 -0.19724 Cos[\[Delta]]-0.0525858 Sin[\[Delta]])^2] (3.77625 -6.10864 Cos[\[Delta]]^2+0.16943 Sin[\[Delta]]-4.70718 Sin[\[Delta]]^2+Cos[\[Delta]] (-15.4993+5.25663 Sin[\[Delta]]))

Could someone help, on how to extract the terms or how to solve a polynomial with one or potentially two square roots?

Thanks a lot!

Answer 1

I'm not sure this is what is sought but it might give some ideas.

ee = -1.52537 - 2.54462 Cos[\[Delta]]^3 + 
   Cos[\[Delta]]^2 (2.94197 - 1.35683 Sin[\[Delta]]) + 
   2.44869 Sin[\[Delta]] - 0.853005 Sin[\[Delta]]^2 + 
   Cos[\[Delta]] (4.69792 - 2.41511 Sin[\[Delta]] - 
      0.180872 Sin[\[Delta]]^2) + 
   Sqrt[1/25 - (0.102901 - 0.19724 Cos[\[Delta]] - 
         0.0525858 Sin[\[Delta]])^2] (3.77625 - 
      6.10864 Cos[\[Delta]]^2 + 0.16943 Sin[\[Delta]] - 
      4.70718 Sin[\[Delta]]^2 + 
      Cos[\[Delta]] (-15.4993 + 5.25663 Sin[\[Delta]]));
gb = 
 Expand[N[First[GroebnerBasis[Rationalize[Rationalize[ee], 0]]]]/10^23]

(* Out[51]= 4.76836353514 - 28.6703276402 Cos[\[Delta]] + 
 31.7329485231 Cos[\[Delta]]^2 + 43.5081178499 Cos[\[Delta]]^3 - 
 41.2169708643 Cos[\[Delta]]^4 - 22.8008864264 Cos[\[Delta]]^5 + 
 19.8169955299 Cos[\[Delta]]^6 - 19.1556952231 Sin[\[Delta]] + 
 77.1822425435 Cos[\[Delta]] Sin[\[Delta]] - 
 12.9216330024 Cos[\[Delta]]^2 Sin[\[Delta]] - 
 68.7910291343 Cos[\[Delta]]^3 Sin[\[Delta]] + 
 10.0472952743 Cos[\[Delta]]^4 Sin[\[Delta]] + 
 12.952125884 Cos[\[Delta]]^5 Sin[\[Delta]] + 
 24.171807059 Sin[\[Delta]]^2 - 
 57.15657135 Cos[\[Delta]] Sin[\[Delta]]^2 - 
 35.9372128245 Cos[\[Delta]]^2 Sin[\[Delta]]^2 + 
 24.7902022477 Cos[\[Delta]]^3 Sin[\[Delta]]^2 + 
 12.1118854195 Cos[\[Delta]]^4 Sin[\[Delta]]^2 - 
 9.35651606565 Sin[\[Delta]]^3 + 
 6.28655294722 Cos[\[Delta]] Sin[\[Delta]]^3 + 
 17.0532282882 Cos[\[Delta]]^2 Sin[\[Delta]]^3 + 
 0.385377612365 Cos[\[Delta]]^3 Sin[\[Delta]]^3 - 
 0.0125817869548 Sin[\[Delta]]^4 + 
 0.800114640976 Cos[\[Delta]] Sin[\[Delta]]^4 + 
 0.258959299084 Cos[\[Delta]]^2 Sin[\[Delta]]^4 - 
 0.610513917166 Sin[\[Delta]]^5 + 
 0.806975356565 Cos[\[Delta]] Sin[\[Delta]]^5 + 
 0.153178774605 Sin[\[Delta]]^6 *)

Answer 2

On the simple example, something like this does the work:

CoefficientList[expr, DeleteDuplicates@Cases[expr, Power[_, 1/2], Infinity]]

On the more complicated ones, a better example would be needed for me to test how to extend the above idea.

OP's example:

sqrts = DeleteDuplicates@Cases[expr, Power[_, 1/2], Infinity]
(*  {Sqrt[1/25 - (0.102901 - 0.19724 Cos[δ] - 0.0525858 Sin[δ])^2]}  *)

{a, b} = CoefficientList[expr, sqrts];

a
(*
-1.52537 + 4.69792 Cos[δ] + 2.94197 Cos[δ]^2 - 
 2.54462 Cos[δ]^3 + 2.44869 Sin[δ] - 2.41511 Cos[δ] Sin[δ] - 
 1.35683 Cos[δ]^2 Sin[δ] - 0.853005 Sin[δ]^2 - 0.180872 Cos[δ] Sin[δ]^2
*)

b
(*
3.77625 - 15.4993 Cos[δ] - 6.10864 Cos[δ]^2 + 
 0.16943 Sin[δ] + 5.25663 Cos[δ] Sin[δ] -  4.70718 Sin[δ]^2
*)

Check:

FromDigits[{b, a}, First@sqrts] - expr // Simplify
(*  0.  *)

---

Extension to multiple square roots:

expr2 = a + {b}.sqrts + 2 a Sqrt[6.10864 Cos[δ]^2 + 0.16943 Sin[δ]];

sqrts2 = DeleteDuplicates@Cases[expr2, Power[_, 1/2], Infinity]
(*
{Sqrt[1/25 - (0.102901 - 0.19724 Cos[δ] - 0.0525858 Sin[δ])^2],
 Sqrt[6.10864 Cos[δ]^2 + 0.16943 Sin[δ]]}
*)

c2 = CoefficientList[expr2, sqrts2];
basis = PadLeft[List /@ sqrts2, {Automatic, 2}, 1];

Fold[Dot, c2, Reverse@basis] - expr2 // Simplify
(*  0.  *)

Fold[Dot, Transpose@c2, basis] - expr2 // Simplify
(*  0.  *)

The FromDigits example for reconstructing the polynomial from the docs for CoefficientList needs an awkward tweak:

Fold[ReleaseHold@ FromDigits[Hold /@ Reverse@#1, #2] &, c2, sqrts2] - 
  expr2 // Simplify
(*  0.  *)