Fundamental Theorem of Algebra

Every complex function will be perfectly factorable.

Theorem

1. $p(x)\in P_n(\mathbb{R})$ is a degree $n$ polynomial with Real Number coefficients represented as $p(x)=x^n+a_{n-1}+\cdots +a_0$
2. Then, $p(x)=(x-{\lambda }_1{)}^{m_1}\ast \dots (x-{\lambda }_k{)}^{m_k}$ Where:

• $m_1+\cdots +m_k=n$
• ${\lambda }_i\in \mathbb{C}$

Alternate Definition

Every $f\in \mathbb{C}[x]$ factors completely. In other words,

• $\exists c_1,\dots ,c_k\in \mathbb{C}$
• $\exists r_1,\dots ,r_k\in \mathbb{Z}$ Such that:
• $f=a(x-c_1{)}^{r_1}\dots (x-c_n^{r_n})$