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In this paper, we present some necessary and sufficient conditions underwhich an irreducible polynomial is self-reciprocal (SR) orself-conjugate-reciprocal (SCR). By these characterizations, we obtain someenumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$,$\lambda\in \Bbb F_q^*$, over $\Bbb F_q$, which are just open questions posedby Boripan {\em et al} (2019). We also count the numbers of Euclidean andHermitian LCD constacyclic codes and show some well-known results on Euclideanand Hermitian self-dual constacyclic codes in a simple and direct way.

Self-reciprocal and self-conjugate-reciprocal irreducible factors of $x^n-\lambda$ and their applications