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Given a polynomial like $x^3 + a_2 x^2 + a_1 x + a_0$ with roots $r_i$, I would like to symbolically compute the coefficients of a polynomial whose roots are $r_i^3 + r_i + 1$. How can I do this in Mathematica?

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I upvoted the other responses. That said, there is a better way.

CoefficientList[
 Resultant[x^3 + a2*x^2 + a1*x + a0, y - (x^3 + x + 1), x], y]

(* Out[1179]= {-1 + 4 a0 - 3 a0^2 + a0^3 - a1 - 2 a0 a1 + 2 a1^2 + 
  a0 a1^2 - a1^3 + a2 + a0 a2 - 2 a0^2 a2 - 3 a1 a2 + 3 a0 a1 a2 + 
  a0 a2^2 - a1 a2^2 + a2^3, 
 3 - 6 a0 + 3 a0^2 + a1 - 2 a1^2 + a1^3 - 2 a2 - a0 a2 + 6 a1 a2 - 
  3 a0 a1 a2 + a1 a2^2 - 2 a2^3, -3 + 3 a0 + a2 - 3 a1 a2 + a2^3, 1} *)
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The direct way works:

Times @@ Table[
     x - With[{r = Root[#1^3 + a2 #1^2 + a1 #1 + a0 &, i]}, r^3 + r + 1], 
     {i, 3}] // 
      Expand // 
      Simplify // 
      CoefficientList[#, x] &

{-1 + 4 a0 - 3 a0^2 + a0^3 - a1 - 2 a0 a1 + 2 a1^2 + a0 a1^2 - a1^3 + 
  a2 + a0 a2 - 2 a0^2 a2 - 3 a1 a2 + 3 a0 a1 a2 + a0 a2^2 - a1 a2^2 + 
  a2^3, 

  3 - 6 a0 + 3 a0^2 + a1 - 2 a1^2 + a1^3 - 2 a2 - a0 a2 + 
  6 a1 a2 - 3 a0 a1 a2 + a1 a2^2 - 2 a2^3, 

  -3 + 3 a0 + a2 - 3 a1 a2 + a2^3, 

  1}
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Another direct approach

p1 = #1^3 + a2 #1^2 + a1 #1 + a0 &;
p2 = 1 + # + #^3 &;

Simplify@CoefficientList[Product[x - p2@Root[p1, i], {i, Exponent[p1@x, x]}], x]

Mathematica graphics

Or make it a function that takes two polynomials, p1 and p2, and produces the coefficients of the polynomial p2 evaluated at the roots of p1

ClearAll[f]
f = Simplify@CoefficientList[Product[x - #2@Root[#, i], {i, Exponent[#@x, x]}],
      x] &;

Examples:

f[p1, p2]

Mathematica graphics

f[p1, 2 + #^2 &]

Mathematica graphics

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A slick way is to use the Newton-Girard formulae in conjunction with the handy RootSum[] function:

Solve[Table[s[m] == RootSum[Function[x, x^3 + b x^2 + c x + d], 
                            Function[r, (r^3 + r + 1)^m]], {m, 3}] ~Join~
      Table[-Sum[s[k] e[m - k], {k, m - 1}] - m e[m] == s[m], {m, 3}], 
      Array[e, 3], Array[s, 3]] // Expand
   {{e[1] -> -3 + b + b^3 - 3 b c + 3 d, 
     e[2] -> 3 - 2 b - 2 b^3 + c + 6 b c + b^2 c - 2 c^2 + c^3 - 6 d - b d -
             3 b c d + 3 d^2, 
     e[3] -> -1 + b + b^3 - c - 3 b c - b^2 c + 2 c^2 - c^3 + 4 d + b d + b^2 d - 2 c d +
             3 b c d + c^2 d - 3 d^2 - 2 b d^2 + d^3}}
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