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In code-division multiple-access (CDMA) transmissions, computing the multiuser minimum mean-squared error (MMSE) detector coefficients requires the inversion of the covariance matrix associated to the received vector signal, an operation usually difficult to implement when the spreading factor and the number of users are large. It is therefore interesting to approximate the inverse by a matrix polynomial. In this correspondence, means for computing the polynomial coefficients are proposed in the context of CDMA downlink transmissions on frequency-selective channels, the users having possibly different powers. Derivations are made in the asymptotic regime where the spreading factor and the number of users grow toward infinity at the same rate. Results pertaining to the mathematics of large random matrices, and in particular to free probability theory, are used. Spreading matrices are modeled as isometric random matrices (spreading vectors orthonormality is a natural assumption in downlink) and also as random matrices with independent and identically distributed (i.i.d.) elements.