Let F q be a finite field of odd order q. In this paper, the irreducible factorization of x 2 a p b r c − 1 over F q is given in a very explicit form, where a, b, c are positive integers and p, r are odd prime divisors of q − 1. It is shown that all the irreducible factors of x 2 a p b r c − 1 over F q are either binomials or trinomials. In general, denote by v p (m) the degree of prime p in the standard decomposition of the positive integer m. Suppose that every prime factor of m divides q − 1, one has (1) if v p (m) ≤ v p (q − 1) holds true for every prime number p|q − 1, then every irreducible factor of x m − 1 in F q is a binomial; (2) if q ≡ 3(mod 4), then every irreducible factor of x m − 1 is either a binomial or a trinomial.