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I am trying to understand the Weierstrass factorization theorum and I was told that the following is true for entire functions...

  1. Any finite sequence $\{c_{n}\}$ in the complex plane has an associated polynomial p(z) that has zeroes precisely at the points of that sequence, $p(z)=\prod_{n}(z-c_{n})$.

  2. Also, any polynomial function p(z) in the complex plane has a factorization $p(z)=a\prod _{n}(z-c_{n})$, where a is a non-zero constant and $c_{n}$ are the zeroes of p.

Can anyone explain this to me in Layman's terms in way that I can understand why these two rules are true, or perhaps just provide a toy example so that I can visualize this better. I understand what complex functions are but I can't seem to understand why these rules hold true.

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To explain the first one, just take any finite sequence, call the terms $c_1,c_2,\ldots,c_n$. The only way a polynomial can equal zero at some point $z_0$ is if $(z-z_0)$ is a factor of the polynomial. Well, if we know all of the factors of the polynomial, we can just multiply the factors together. Putting this information together, we want a polynomial that vanishes at $c_1,\ldots, c_n$ and nowhere else; well, $z-c_1$ must be a factor, as must $z-c_2$, and so on and so forth. Thus, up to a multiplicative constant, our polynomial must equal $$P(z)=a(z-c_1)(z-c_2)\cdots(z-c_n)=a\prod_{k=1}^n(z-c_k).$$ We need the constant in front because, as long as $a\neq0$, the polynomial is still going to have the same zeroes and only these zeroes. Once you impose a condition, such as $P(1)=6$, then you can determine an explicit $a$.

Your second question is precisely the fundamental theorem of algebra; I think it is most easily expressed as the following:

A polynomial of degree $n$ has exactly $n$ complex zeros.

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