Students first encounter the term polynomial in the third year of secondary school; however, they develop its concept at an earlier stage. In secondary technical school, we study polynomials as real functions, counting their zeros, exploring their behavior at infinity and drawing their graphs. At the university level, on the other hand, polynomials are considered from a slightly different perspective, by defining them as infinite sequences of elements of a ring K. In the theoretical part of the master's thesis, the definition and basic characteristics of rings are introduced. This lays the basis for introducing the ring of the polynomials with coefficients in a given ring, since some properties of the coefficient ring $K$ are transferred to the ring of polynomials K[x]. Later on, we introduce polynomial evaluation at a point, which will be used to define polynomial zeros. The division algorithm and some of its corollaries are recalled. Moreover, we compare this abstract content to its discussion and consideration in primary and secondary school, where these topics are presented in a simplified way and are restricted to polynomials with real coefficients. The focus of our observation is the connection between the algebraic and the graphical representation of polynomials. The empirical part is a practical investigation comparing two different approaches to teaching graphing polynomial functions. The first one is a classical treatment of the topic, while the second one is learning by independent discovery using the program Desmos.
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