MathDB

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)

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

Problem Statement