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It is shown how the use of relatively sparse polynomials, including p-polynomials and trace polynomials, can be used as intermediate divisors in the Goertzel transform over a finite field to reduce the number of additions. The number of multiplications can also be reduced if the characteristic of the field is larger than two. These methods can also be used in preliminary stages of a finite-field Winograd transform. Applications are for the decoding of Reed-Solomon and Bose-Chaudhuri-Hocquenghen codes in the spectral mode.