On one hand polynomials are defined by the most straightforward operations. One doesn't need to involve inequalities (nor division). In this sense polynomials form a narrow class about which it is easy to prove useful theorems and algorithms. On the other hand, as @JO'R has mentioned in the Question, all continuous functions are approximated by polynomials. Thus this family is in this sense rich and useful in applications.
Włodzimierz Holsztyński
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