  We study the explicit factorization of $2^n r$-th cyclotomic polynomials over
finite field $\mathbb{F}_q$ where $q, r$ are odd with $(r, q) =1$. We show that
all irreducible factors of $2^n r$-th cyclotomic polynomials can be obtained
easily from irreducible factors of cyclotomic polynomials of small orders. In
particular, we obtain the explicit factorization of $2^n 5$-th cyclotomic
polynomials over finite fields and construct several classes of irreducible
polynomials of degree $2^{n-2}$ with fewer than 5 terms. The reciprocals of
these irreducible polynomials are irreducible polynomials of the form
$x^{2^{n-2}} + g(x)$ such that the degree of $g(x)$ is small ($\leq 4$), which
could have potential applications as mentioned by Gao, Howell, and Panario in
\cite{GaoHowellPanario}.
