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 \(\deg A \ge \deg B\).

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

\(\deg(Q) = \deg(A) - \deg(B)\)
\(\deg(R) \lt \deg(B)\) 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 \(\deg = 1\). 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