This text presents easy to understand proofs of some of the most difficult results about polynomials. It encompasses a self-contained account of the properties of polynomials as anlytic functions of a special kind. The zeros of compositions of polynomials are also investigated along with their growth and some of these considerations lead to the study of analogous questions for trigonometric polynomials and certain transcendental entire functions. The strength of methods are fully explained and demonstrated by means of applications.