This paper belongs to the class of iterative equations which are of special interest for people in dynamical systems and functional equation. Precisely, the authors deal with the polynomial-like iterative equation
where are real numbers. The case , , with as unknown, is studied with detail. Some previous results are extended and the case of multiple characteristic roots is faced. They describe such equation in linear difference form, give relations between the characteristic solutions and other solutions and they construct in specific cases all continuous solutions. It is shown that the equation has no continuous real solutions of it has no real characteristic roots.