Question 125
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f(0)=2004$.
2. If $x$ is irrational, then $f(x)$ is also irrational.
(Notes: A polynomial is monic if its highest degree term has coefficient 1. Thus, $f(x)=x^{4}-5 x^{3}-4 x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2 / 5$ and $-9$ are rational, while $\sqrt{2}$ and $\pi$ are well known to be irrational.)
1. $f(0)=2004$.
2. If $x$ is irrational, then $f(x)$ is also irrational.
(Notes: A polynomial is monic if its highest degree term has coefficient 1. Thus, $f(x)=x^{4}-5 x^{3}-4 x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2 / 5$ and $-9$ are rational, while $\sqrt{2}$ and $\pi$ are well known to be irrational.)