  The well-known mathematical instrument for detection common roots for pairs
of polynomials and multiple roots of polynomials are resultants and
discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$
is a function of their coefficients. Zeros of resultant $R(f,g)$ correspond to
the families of coefficients of $f$ and $g$ such that $f$ and $g$ have a common
root. Herewith the calculation of this common root is a separate problem.
  The principal results on calculation of a unique common root of two
polynomials and also about calculating a unique root of multiplicity 2 of a
polynomial in terms of the first order partial derivatives of resultants and
discriminants are given in the monograph by I.M. Gelfand, M.M. Kapranov, A.V.
Zelevinsky [1, Ch. 3, Ch. 12]. A significant development of the ideas of this
book in the direction of searching for formulas for multiple roots of
polynomials is presented in the paper by I.A. Antipova, E.N. Mikhalkin, A.K.
Tsikh [2]. The key result of this article is [2, Theorem 1] where the
expression for a unique root of multiplicity $s \geq 3$ in terms of the first
order partial derivatives of resultant of the polynomial and it's derivative of
order $s-1$.
  In the present article the explicit formulas for higher derivatives of
resultants of pairs of polynomials possessing common roots are obtained. On
this basis a series of results that differ in ideas from [2, Theorem 1] linking
higher derivatives of resultants and common multiple roots are proven. In
addition the results obtained are applied for a new transparent proof of a
refinement of [2, Theorem 1].
