examday

/TMUA

How many monic polynomials $p$ of degree $6$ with integer coefficients satisfy all of the following conditions?

• The coefficient of $x^5$ is $0$ and the coefficient of $x^4$ is $-30$.
• For infinitely many positive integers $n$, the number $p(n)$ is the square of an integer.
• The polynomial $p$ has exactly three distinct real zeros, none of which is an integer.