I Polynomial of finite degree actually infinite degree?

[Stephen Tashi]

$1+x+x^2 = \dfrac{1-x^3}{1-x} = (1-x^3)\cdot \dfrac{1}{1-x} = (1-x^3)\sum_{k=0}^\infty x^k$.

Isn't this a contradiction since the LHS has degree $2$ while the RHS has infinite degree?

By the usual definition of "polynomial" the RHS is not a polynomial. (The usual definition of polynomial doesn't permit a "polynomial" to be specified by the operation of taking a limit. Do your course materials actually define a type of "polynomial" that has "infinite degree"? )

A polynomial function f(x) may be equal to a function that is expressed in a way not permitted in the definition of "polynomial". For example f(x) = x = x + sin(x) - sin(x).

What qualifies a function to be a polynomial is that it may be expressed in a certain way. Alternative ways of expressing it don't involve any logical contradiction.