Suppose we have a polynomial $p$
$$ p = a_0 + a_1x + a_2x^2 + \dots+a_Nx^N = \sum_{i=0}^Na_ix^i. $$
what is the sufficient and necessary condition of $\{a_i:i=0,1,\dots, N\}$ such that $p$ is always positive for all $x\in(0, +\infty)$
(does such sufficient and necessary conditions even exist?)
In other words: what kind of (and necessary) coefficients could make the polynomial always positive on the right side of $\mathbb{R}?$
Guess: what about let $\xi_k$ be all of the roots of $\frac{dp}{dx}=0$, and the condition is $p(\xi_k)$ are all positive, and $a_N>0$?
A necessary and sufficient condition uses Sturm's theorem. You want $a_N > 0$ and $V(0) = V(\infty)$, where $V(x)$ is the number of sign changes of the Sturm sequence of $p$ at $x$.