# Root of symbolic high-degree polynomial

I have a symbolic equation EQ, expressed in terms of symbolic variables d, f, t. The equation contains a number of 6th degree polynomial roots (with a symbolic coefficient t). The equation is quite long and it starts like this:

EQ = ( d f* (8root(z^6 – 12tz^5 + (243t^2z^4)/8 + (399t^3z^3)/4 – (837t^4z^2)/2 – (639t^5z)/4 + (9945t^6)/8, z, 1)^5 + 155t^2root(z^6 – 12tz^5 + (243t^2z^4)/8 + (399t^3z^3)/4 – (837t^4z^2)/2 – (639t^5z)/4 + (9945t^6)/8, z, 1)^3 + …. and so on

Matlab is unable to solve it with the solve() function. My initial thought was that the explicit solution of a polynomial with arbitrary coefficients does not exist (Abel-Ruffini theorem), but I evaluated the expression for a set of numeric variables and then by running regression and guessing the form of the solution I was able to find the exact solution (expressed with symbolic variables d, f, t):

EQ= RegressionConstant * t * d * t

This puzzles me. If the exact solution does exist, why would Matlab not give it to me straight away? What am I missing?

Thanks for any suggestions

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