TITLE: When does a polynomial have only nonnegative real roots
QUESTION [9 upvotes]: Is there a general criterion to determine whether a polynomial with rational coefficients has the property that all of its roots are real and nonnegative?

REPLY [2 votes]: The full answer is a little complicated to type out. I suggest looking at Sturm's Theorem. The Sturm sequence of a polynomial $P(x)$ can be used  to determine the number of zeros of $P(x)$ in a given interval. It is a crucial element in Tarski's decision procedure for the truth in the reals of sentences of the elementary theory of fields.