In the present paper we give all complete decompositions over \(\mathbb {C}\) of the polynomial \(\begin{aligned} P(x)=(x-a_1)^{k_1}\cdots (x-a_m)^{k_m}, \end{aligned}\) where \(2\le m\) , \(m\notin \left\{ 4, 6, 7, 10\right\} \) is a positive integer, \(a_1, a_2, \ldots , a_m\) are pairwise distinct real numbers, \(k_1,\) \(k_2,\) \(\ldots ,\) \(k_m\) are positive integers and the number of pairwise distinct integers among \(k_1, k_2,\ldots , k_m\) is greater than \((m+1)/2\) . As a consequence, we prove that if \(a_1, a_2, \ldots , a_m\) are pairwise distinct rational numbers and \(f(x)\in \mathbb {Q}[x]\) is an irreducible polynomial over \(\mathbb {Q}\) , then under certain conditions related to the exponents \(k_1, k_2, \ldots , k_m\) and the degree of the polynomial f(x), the polynomial \(\begin{aligned} P(x)=f(x)^n(x-a_1)^{k_1}\cdots (x-a_m)^{k_m} \end{aligned}\) is always indecomposable over the field of complex numbers.
