The fundamental theorem of algebra in which we prove a complex polynomial has at least root is clear from the construction of a compact domain and use of the polar coordinate form of complex numbers. However, what I do not understand is how the fundamental theorem of algebra can be extended to show the number of roots and not just the existence of a complex root?
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In short, the division algorithm tells you this. Once you have a root $\alpha$, you divide your polynomial $f(x)$ by $x-\alpha$ and you get a polynomial of smaller degree. So existence of roots and induction with the division algorithm give you the correct number of roots counted with multiplicity.
Dylan Yott
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By the division algorithm, we can write
$$f(z) = (z - a) q(z) + r(z)$$
with $q, r$ polynomials. Now $a$ is a root of $f$ if and only if the remainder $r$ is zero; it's immediate that $\deg q = \deg f - 1$, and induction can be used.