The zero-free intervals for characteristic polynomials of matroids

Let M be a loopless matroid with rank r and c components. Let P(M, t) be the characteristic polynomial of M. We shall show that (-1)P-r (M,t) greater than or equal to (1 - t)(r) for t is an element of (-infinity, 1), that the multiplicity of the zeros of P(M, t) at t = 1 is equal to c, and that (1)Pr+c(M, t) greater than or equal to (t - 1)(r) for t is an element of (1, 32/27]. Using a result of C. Thomassen we deduce that the maximal zero-free intervals for characteristic polynomials of loopless matroids are precisely (-infinity, 1) and (1, 32/27].