TITLE: Zeros of $C^\infty$ functions
QUESTION [6 upvotes]: If $f(x)  \in C^\infty(\Bbb{R})$,and $f(a)=0$, do we have $$f(x)=(x-a)g(x)$$? where $g(x) \in C^\infty(\Bbb{R})$ and $g(a)=f'(a)$

REPLY [6 votes]: For $a=0$ (to simplify things): $f(x)=\int_0^1\frac{d}{dt}f(tx)dt=x\int_0^1f'(tx)dt$, i.e. $g(x)=\int_0^1 f'(tx)dt$.