This paper presents majority-decoding algorithms for four classes of binary cyclic codes. The classes are those for which the parity-check polynomial is: 1) the product of two primitive polynomials with relatively prime exponents and 2) the product of(x^r + 1)/(x + 1)and a primitive polynomial, wherer \geq 3is odd and the exponents are relatively prime, together with the corresponding nonexpurgated codes.