  In this paper we settle some polynomial identity which provides a family of
explicit Waring decompositions of any monomial $X_0^{a_0}X_1^{a_1}\cdots
X_n^{a_n}$ over a field $\Bbbk$. This gives an upper bound for the Waring rank
of a given monomial and naturally leads to an explicit Waring decomposition of
any homogeneous form and, eventually, of any polynomial via (de)homogenization.
Note that such decomposition is very useful in many applications dealing with
polynomial computations, symmetric tensor problems and so on. We discuss some
computational aspect of our result as comparing with other known methods and
also present a computer implementation for potential use in the end.
