Simplify an expression

Question

exp = (1 + x)^30 + (1 - x)^30;
Factor[exp]
    (*2 (1 + x^2) (1 + 14 x^2 + x^4) (1 + 44 x^2 + 166 x^4 + 44 x^6 + 
   x^8) (1 + 376 x^2 + 4380 x^4 + 15944 x^6 + 24134 x^8 + 
   15944 x^10 + 4380 x^12 + 376 x^14 + x^16)*)
FullSimplify[%]
(*returns above factorized expression*)
Expand[%]//FullSimplify
(*returns above factorized expression*)

Any ideas how to instruct Mathematica to get the original exp expression? Thanks!

Answer 1

Clear["Global`*"]

exp = (1 + x)^30 + (1 - x)^30;

exp2 = Factor[exp]

(* 2 (1 + x^2) (1 + 14 x^2 + x^4) (1 + 44 x^2 + 166 x^4 + 44 x^6 + x^8) (1 + 
   376 x^2 + 4380 x^4 + 15944 x^6 + 24134 x^8 + 15944 x^10 + 4380 x^12 + 
   376 x^14 + x^16) *)

If you assume that exp2 can be expressed in the form (a*x+b)^30+(c*x+d)^30

(sol = (a*x + b)^30 + (c*x + d)^30 /. 
      Solve[Thread[
        CoefficientList[exp2, x] == 
         CoefficientList[(a*x + b)^30 + (c*x + d)^30, x]], {a, b, c, 
        d}, Reals] // Simplify // Union)[[1]]

(* (-1 + x)^30 + (1 + x)^30 *)