Is $ \frac{x^2 -4}{x-2} $ a polynomial? The expression $ \frac{x^2 -4}{x-2}$ doesn’t looks like a polynomial because of $ x $ in the denominator. But it can be factorized. After factorizing and canceling the common factor $$\frac{x^2 -4}{x-2} = \frac{(x+2)(x-2)}{x-2} = x+2, $$ and $ x+2 $ is a polynomial which satisfies the definition of the polynomial.
Is the original expression $ \frac{x^2 -4}{x-2} $ is a polynomial or not?

It is not because it is not equivalent to $x+2$.  You can evaluate $x+2$ for any real $x$ if you are working in the reals.  You cannot evaluate $\frac {x^2-4}{x-2}$ at $x=2$ because of the division by $0$
