Let F(x, y) (￠) (R)[x, y] a polynomial function, p(x) (￠) (R)[x] an irreducible polynomial function and m (￠)( N), a positive integer. In this paper, we will find a polynomial solution y = y(x) (￠) (R)[x] to satisfy the polynomial equation: F(x, y(x)) = g(x)pm(x) (*) for some polynomial function g(x)(￠) (R)[x]. The main result of this paper is to find an upper bound of the number for all solutions y(x) for the extended polynomial problem (*) if the number of solutions is finitely many.
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