examday

/TMUA

The polynomial

has integer coefficients, and both $f(0)$ and $f(1)$ are odd integers.

You may use the fact that any rational root of a polynomial with integer coefficients and leading coefficient $1$ is an integer.

Consider the statements

• I. $f(n)$ is odd for every integer $n$.
• II. Every real root of $f(x)=0$ is irrational.
• III. The equation $f(x)=0$ has exactly one real root.

Which of these statements must be true?