The notion of a root functional of a system of polynomials or an ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A basis of the space of all root functional and a basis of the ideal are found by using the operation of extension of bounded root functionals when the number of equations is equal to the number of unknowns and if it is known that the number of roots is finite. The asymptotic complexity of these methods is <$E d sup O(n)> operations, where <$En> is the number of equations and unknowns and <$Ed> is the maximal degree of polynomials.

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