  We use the elimination theory to explicitly construct the (n-3)! order
polynomial in one of the variables of the scattering equations. The answer can
be given either in terms of a determinant of Sylvester type of dimension (n-3)!
or a determinant of B\'ezout type of dimension (n-4)!. We present a recursive
formula for the Sylvester determinant. Expansion of the determinants yields
expressions in terms of Pl\"ucker coordinates. Elimination of the rest of the
variables of the scattering equations is also presented.
