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This paper introduces a novel algorithm to estimate the direction-of-arrival (DOA) and the polarization of a completely-polarized polynomial-phase signal of an arbitrary degree. The algorithm utilizes a polarized vector-sensor, comprising a spatially collocated six-component electromagnetic vector-sensor, a dipole triad, or a loop triad. This ESPRIT-based algorithm is based on a time-invariant matrix-pencil pair, derived from the time-delayed data-sets collected by a single polarized vector-sensor. The high-order difference-function of the signal's phase constructs the invariant-factor used in the ESPRIT algorithm. The steering vector is estimated from the signal-subspace eigenvector of the data-correlation matrix, following which the closed-form DOA and polarization can be obtained. Given the degree of the polynomial-phase signal, this approach resolves the two-dimensional azimuth-elevation angle and the polarization of the source, and requires neither a priori knowledge of the polynomial-phase signal's coefficients nor a priori knowledge of the polynomial-phase signal's frequency-spectrum. The efficacy of the proposed algorithm is verified by Monte Carlo simulations. Estimation accuracies of the DOA and the polarization parameters are evaluated by the closed-form Cramér-Rao bounds, which are independent of the polynomial coefficients, the degree of the polynomial-phase signal, and the azimuth-angle of the source.