Determine how many roots of special n-degree polynomial are positive integers

Question

I have a special polynomial of the form $x^n + bx + c = 0$
$c = b - 1$
How can I determine how many of the possible values of $x$ are positive integers?
All values of $n$ will be positive integers.

Answer 1

You can combine the rational root test with Descartes' rule of signs to obtain an upper bound for the number of positive (real) roots.

If $n$ is fixed, Sturm's algorithm will give you the exact number of (real) positive roots $\le c$.