Multiplicity of a root

Builds on Root of a polynomial Degree of a polynomial

Definition

Let $R$ be a commutative ring, let $f \in R[x]$ with $f \neq 0$ and let $c \in R$ . The multiplicity of $c$ as a root of $f$ is the largest integer $m \ge 0$ such that $(x - c)^m$ divides $f$ .

Remarks

The integer $m$ exists, because $(x - c)^m$ is monic of degree $m$ , so it cannot divide $f$ for $m > \deg f$ . The multiplicity is positive exactly when $c$ is a root of $f$ . A root of multiplicity $1$ is a simple root. A root of multiplicity at least $2$ is a repeated root. In $\mathbb{Q}[x]$ , $f = (x - 1)^3 (x + 2)$ has the root $1$ with multiplicity $3$ and the root $-2$ with multiplicity $1$ .

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Group concept
Ring definition
Polynomial ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Root of a polynomial definition
Multiplicity of a root definition

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16 pages build on this one.

Tag	`03XVP8Q4GY`
URI	`m2://03XVP8Q4GY`
Kind	definition
Writing	W0 stub
MSC	13F20
Defines	Multiplicity of a root
Uses	Root of a polynomial, Degree of a polynomial

Multiplicity of a root

Definition

Remarks

Reading path

Where this leads

Further reading

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