It is well known that a polynomial of degree $n$ admits $n$ roots when the field is algebraically closed. However, this comes with a caveat, in particular that the roots be counted with multiplicity.
From an algebraic standpoint, counting roots with multiplicity is very natural: If a root $\alpha_i$ has multiplicity $m_i$, then we may factorize the polynomial as $a \displaystyle \prod_{i=1}^l (x - \alpha_i)^{m_i}$.
However, this is unsatisfying to me since we are first introduced to roots of a polynomial as a geometric concept: in the real case, it's places where the curve crosses the $x$-axis. Thus, when we look at the theorem from a purely geometric standpoint without looking at the underlying algebraic framework, the theorem becomes a lot less interesting: the polynomial of degree $n$ admits $n$ roots, but it can seem to admit less if some of those roots happen to have multiplicity greater than one.
My question is, is there any geometric relation between a root and its multiplicity which allow us to see the full strength of the theorem (the existence of $n$ roots) without relying on the underlying algebraic structure of the polynomial? Stated differently, can we look at the graph/image of a polynomial and determine how many roots it has (without appealing to arguments about its degree and inferring the number of roots from that)?
Note that my question can also be applied to something like Bezout's theorem where plane curves of degree $m$, $n$ intersect $mn$ times, assuming the intersections are counted with multiplicity. The condition that they be counted with multiplicity is even more disappointing to the geometric nature here.
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