ILL 1987 Problem 39

Let $A$ be a set of polynomials with real coefficients and let them satisfy the following conditions:(i) if $f \in A$ and $\deg( f ) \leq 1$ , then $f(x) = x - 1$ ;(ii) if $f \in A$ and $\deg( f ) \geq 2$ , then either there exists $g \in A$ such that $f(x) = x^{2+\deg(g)} + xg(x) -1$ or there exist $g, h \in A$ such that $f(x) = x^{1+\deg(g)}g(x) + h(x)$ ;(iii) for every $g, h \in A$ , both $x^{2+\deg(g)} + xg(x) -1$ and $x^{1+\deg(g)}g(x) + h(x)$ belong to $A.$ Let $R_n(f)$ be the remainder of the Euclidean division of the polynomial $f(x)$ by $x^n$ . Prove that for all $f \in A$ and for all natural numbers $n \geq 1$ we have $R_n(f)(1) \leq 0$ , and that if $R_n(f)(1) = 0$ then $R_n(f) \in A$ .