TITLE: Proof of how multiplicity in a polynomial works.
QUESTION [8 upvotes]: In Algebra (2) I was told that if a polynomial had an even multiplicity for some $x=a$, then the graph touches $y=0$ at $x=a$ but doesn't cross $y=0$.  Odd multiplicities go through the $x$-intercept.  For example:$$y=x^2\to y=(x-0)(x-0)\to x=0,0$$And you can clearly see the graph "touches without intersecting" at $x=0$.
However, I am confused on how this is proven.

REPLY [5 votes]: Let $P(x) = Q(x)(x-a)^{2n}$ such that $(x-a)\nmid Q(x)$. In a small enough neighbourhood of $a$ (for instance one that contains no roots of $Q(x)$), then $Q(x)$ preserves sign. And $(x-a)^{2n}\geq 0$, therefore, in  said neighbourhood, $P(x)$ preserves sign, i.e., the function does not cross the line $y=0$.
You can find a similar argument of why it happens the other way around for $P(x) =   Q(x)(x-a)^{2n+1}$.