Hyperbolic polynomials are coordinate free generalization to the notion of real rooted polynomials. A special class of hyperbolic polynomials are determinantal polynomials and they bound spectrahedra, feasible sets of semidefinite programming. I shall discuss some techniques of real algebraic geometry to deal with convex semialgebraic sets such as spectrahedra, and  demonstrate the applications of hyperbolic polynomials in combinatorics and optimization. More generally, the overarching idea of using generating polynomials to study mathematical objects like matrix, graph, matroid, known as polynomial paradigm will be emphasized throughout the talk.