MATLAB: Different answers from ‘roots’ & ‘solve’ command

Question

>> R=roots([3 0 0 0 0 4 1])
R =
     0.9017 + 0.6256i
     0.9017 - 0.6256i
    -0.2767 + 1.0147i
    -0.2767 - 1.0147i
    -1.0000 + 0.0000i
    -0.2502 + 0.0000i
>> R(R == real(R))
ans =
     -1.0000
     -0.2502
>> solx = solve(3*x^6 + 4*x + 1==0, x)
solx =
                                              -1
   RootOf(z^5 - z^4 + z^3 - z^2 + z + 1/3, z)[1]
   RootOf(z^5 - z^4 + z^3 - z^2 + z + 1/3, z)[2]
   RootOf(z^5 - z^4 + z^3 - z^2 + z + 1/3, z)[3]
   RootOf(z^5 - z^4 + z^3 - z^2 + z + 1/3, z)[4]
   RootOf(z^5 - z^4 + z^3 - z^2 + z + 1/3, z)[5]
>> solx(solx == real(solx))
ans =
-1
>>

Answer 1

SOLVE was not able to find a closed symbolic form for all the roots (because one doesn't exist for this polynomial). It therefore cannot assess whether the ones not in closed form are real.