:PROPERTIES: :ID: 0d2bae2f-5054-4aba-b1dc-ca704476764f :mtime: 20231026010930 20231019014037 :ctime: 20231019013950 :END: #+title: fundamental theorem of algebra #+filetags: :public:project: * Statement of FTA Given a polynomial on the [[id:2553e0fb-12c1-42cc-8e47-54937a36e2c7][complex numbers]] \[p : \mathbb{C} \to \mathbb{C}\] \[p(x) = a_{0} + a_{1}x + a_{2}x^{2} + \cdots + a_{n}x^{n}\] where $a_0, a_1, a_2 \ldots , a_{n} \in \mathbb{C}$, and $a_n \neq 0$, then the equation \[p(x) = 0\] has exactly $n$ solutions (counted with multiplicity) over the complex numbers. ** Alternative Statement of FTA Another way of saying the same thing is that $p(x)$ can be written as \[p(x) = (x - r_{1}) (x - r_{2}) \cdots (x - r_{n})\] For some $r_1 , r_2 , \ldots r_n \in \mathbb{C}$