The aim of this paper is to study the concept of a polynomial, to describe its algebraic structure and to show the operations and properties on the structure of a polynomial ring. The work is a three-part structure. The first part will recall the basic concepts without whose knowledge the work cannot be understood and which will be used in the work as known facts. The second part of the paper will be based on the polynomials of one variable and observe their divisibility, algorithms for dividing polynomials and determining the greatest common divisor as well as finding and determining the nature of the roots of algebraic equations associated with polynomials. The criteria for reducibility and irreducibility of polynomials and algorithms for solving algebraic equations of the third and fourth degree in radicals will be shown and applied by example. The last part of the paper will briefly look at the representation of polynomials in school education.