Complex Polynomials

Definition: Complex Polynomial

A complex polynomial is a polynomial over the complex numbers.

Polynomial Division

Theorem: Polynomial Division

Let $A(z)$ and $B(z)$ be two complex polynomials such that $\mathrm{deg}A\ge \mathrm{deg}B$.

If $B(z)$ is non-zero, then there exist unique polynomials $Q(z)$ and $R(z)$ such that

$$A(z)=Q(z)B(z)+R(z),$$

where

• $\mathrm{deg}(Q)=\mathrm{deg}(A)-\mathrm{deg}(B)$
• $\mathrm{deg}(R)<\mathrm{deg}(B)$ or $R(z)$ is zero.

We call $A$ the dividend, $B$ the divisor, $Q$ the quotient and $R$ the remainder.

PROOF

TODO

Definition: Divisibility

If $R(z)=0$, then we say that $A$ is divisible by $B$.

The Fundamental Theorem of Algebra

Every non-constant complex polynomial $A(z)={\sum }_{k=0}^n{a}_k{z}^k$ can be factorized into a product of complex polynomials with $\mathrm{deg}=1$. More specifically,

$$A(z)=a_n(z-z_1{)}^{m_1}\cdots (z-z_l{)}^{m_l},$$

where $z_i$ are the distinct roots of $A(z)=0$ and $m_i$ are their respective multiplicities.

PROOF

TODO