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Core Concept Map Language

Corollaries from Complex polynomials

All corollaries from Complex polynomials.

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[COROLLARY]

Count roots with multiplicity

A complex polynomial of degree $n\ge1$ has exactly $n$ complex roots counted with multiplicity.

The factorisation theorem gives $n$ linear factors. Conversely, if $p(\beta)=0$ , then

0=a(\beta-\alpha_1)\cdots(\beta-\alpha_n),

so $\beta$ equals one of the $\alpha_j$ . Thus the list has no missing roots, and repeated factors record multiplicity.

“Exactly $n$ roots” does not mean $n$ distinct values: $(z-1)^2$ has two roots counted with multiplicity, but only one distinct root.

[1]:Factorisation over the complex numbers

This page builds on Complex arithmetic and Polar form.

It leads to Seen questions and Stability of systems.

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Count roots with multiplicity