Is concept of Repeated root only for Polynomials

Question

We know that if a polynomial $f(x)=0$ has repeated root $\alpha$ Then at $x=\alpha$, $x$ axis will be the tangent since $f'(\alpha)=0$

Now can say if at $x=\alpha$ if $x$ axis is the tangent, Then $f(x)$ has repeated root?

Also is this repeated root concept valid for other than polynomial functions?

Answer 1

We use the same term when a Taylor polynomial of a general $f$ centered at $\alpha$ has a multiple root in the sense of polynomials. For example, $f(x)=\sin(x)^2$ has a double root at $x=0$.

Answer 2

People call this the 'vanishing order' when the function is smooth. It's a concept I see mostly in connection to approximation theorems, like Taylor's theorem, and certain other local forms question.

You can find more information here: Wikipedia on zeros and poles of a function.