In the study of the distribution of zeros of polynomials and entire functions the techniques used, roughly speaking, fall into three categories" analytic, geometric and algebraic. In this paper, which represents the first portion of a two-part investigation, we will attempt to exploit the advantages of all three techniques. In Section 2 we will introduce a novel geometric tool (see also [3]) to prove results which, for the most part, are intractable by algebraic or analytic methods. In addition, the geometric theorems are generally stronger than their algebraic counterparts which are derived as corollaries. In the extensive literature dealing with the location of zeros of real polynomials (and real entire functions) a significant role is played by linear transformations T which possess the following property: (1.1) Zc(T[fl) < Zc(f),