The Fundamental Theorem of Algebra.

    

This theorem is also called the theorem of d'Alembert.

Theorem 6.3.1   Let $ P(z)$ be a non constant polynomial over $ \mathbb{C}$ . Then $ P(z)$ has a root.

Corollary 6.3.2   Let $ P(z)$ be a non constant polynomial of degree $ n$ over $ \mathbb{C}$ . Then $ P(z)$ has exactly $ n$ roots, counted with multiplicity.

First examples are displyed in subsection 7.

Corollary 6.3.3   Every non constant polynomial with real coefficients is the product of factors of degree 1 and 2.

Proof. Let $ P(z)$ be a polynomial with real coefficients. By thm 3.1, this polynomial has at least one root $ z_1$ . If this root is real, then $ P(z)$ factors by $ (z-z_1)$ .

Suppose that $ z_1$ is not real. By thm 7.3, $ \overline{z_1}$ is also a root of $ P(z)$ . Thus, $ P(z)$ factors by $ (z-z_1)(z-\overline{z_1})=(z^2-2$   Re$ (z_1)+\vert z_1\vert^2)$ . $ \qedsymbol$

An example can be found in 7.6.

Noah Dana-Picard 2007-12-24