A central theme in algebraic, geometric, and topological combinatorics is the investigation of distributional properties of combinatorial generating polynomials, such as symmetry, log-concavity, and unimodality.  Recently, new questions in the field ask when such polynomials possess another distributional property, called the alternatingly increasing property, which implies unimodality.  The alternatingly increasing property for a given polynomial is equivalent to a unique pair of symmetric polynomials both being unimodal with nonnegative coefficients.  We will discuss a systematic approach to proving the alternatingly increasing property using real zeros of these symmetric polynomials.  We will then look at some applications of these methods to recent questions and conjectures in algebraic combinatorics​