regel = {(α^n)_ :> 
            2  (α^(n - 2)) /; 
              n ≥ 2, (β^m)_ :> 
            5  (β^(m - 3)) /; m ≥ 3};

F = (∑_(k = 0)^6) Expand[(Subscript[c, k]) (((α + β))^k)] //. 
        regel

sol = 
      Solve[Flatten[CoefficientList[F, {α, β}]] == 0, 
        Table[(Subscript[c, k]), {k, 0, 6}]]

F = (∑_(k = 0)^6) (Subscript[c, k]) (x^k) /. sol[[1]] /. 
        (Subscript[c, 6]) -> 1

Factor[F]

\(\(α = Root[#\^2 - 2 &, 2];\)\)
\(\(β = Root[#\^3 - 5 &, 1];\)\)

result = α + β // RootReduce

minpol = result[[1]][x]

α*β // RootReduce

(1/(α + β)) // RootReduce