The Tutte-Gr\"othendieck polynomial T(G;xy) of a graph Gencodes numerous interesting combinatorial quantities associatedwith the graph. Its evaluation in various points in the (xy) plane give the number of spanning forests of the graph, the numberof its strongly connected orientations, the number of its properk-colorings, the (all terminal) reliability probability of thegraph, and various other invariants the exact computation of eachof which is well known to be #P-hard. Here we develop a generaltechnique that supplies fully polynomial randomised approximationschemes for approximating the value of T(G;xy) for any densegraph G, that is, any graph on n vertices whose minimum degree is (n), wheneverx1 and 1">y1, and in various additional points. Annan \cite{1} has dealt with the case y=1, x1.This region includes evaluations of reliability and partition functions of the ferromagnetic Q-state Potts model. Extensions to linear matroids where T specialises to the weight enumerator of linear codes are considered as well.