Remainder theorem

Builds on Division algorithm for polynomials Polynomial evaluation

Statement

Let $R$ be a commutative ring, let $f \in R[x]$ and let $c \in R$ . Then the remainder of the division of $f$ by $x - c$ is the constant polynomial $f(c)$ . That is, there is $q \in R[x]$ with $f = (x - c) q + f(c)$ .

Remarks

The proof: division by the monic polynomial $x - c$ gives $f = (x - c) q + r$ with $r$ a constant. Evaluation at $c$ gives $r = f(c)$ . For example, the remainder of $x^3 - 2x + 5$ divided by $x - 2$ is $2^3 - 4 + 5 = 9$ .

Reading path

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Group concept
Ring definition
Polynomial ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Leading coefficient definition
Monic polynomial definition
Division algorithm for polynomials statement
Remainder theorem statement

Where this leads

10 pages build on this one.

Tag	`0TEDQH56RW`
URI	`m2://0TEDQH56RW`
Aliases	`polynomial-remainder-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	13F20
Status	This copy of the site has no build data. Run `m2 status 0TEDQH56RW`.
Uses	Division algorithm for polynomials, Polynomial evaluation
Same as	Wikidata Q1071948 Wikipedia Polynomial remainder theorem

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Used by

Factor theorem