On 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.

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.