Ivan Shishkin, Birch Grove

Polynomial

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Revision 805

8/1/2026, 5:13:48 PM · Sequoia

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Recorded titlePolynomial
Recorded typeDefinition
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1A polynomial $P$ over a [[Commutative ring|commutative ring]] $A$ is a finite sequence $(a_i)_i$ in $A$. Most of the time, we write it this way
1A polynomial $P$ over a [[Commutative ring|commutative ring]] $A$ is a finite [[Sequence|sequence]] $(a_i)_i$ in $A$. Most of the time, we write it this way
2$$P=\sum_{i=0}^n a_i X^i=a_0+a_1 X+a_2 X^2+\cdots+a_n X^n$$
3where $X$ is called a "formal variable" and $n$ is the biggest integer $k$ such that $a_{k}\neq 0$.
4
5##### Remarks
6- $n$ is called the degree of $P$.
7- The set of all polynomials in $X$ with coefficients in $A$ is denoted $A[X]$.

Revision 804

8/1/2026, 5:13:32 PM · Sequoia

Concept edited

Compare with revision 8024 changed lines
1A polynomial over a [[Commutative ring|commutative ring]] $A$ is a formal expression
1A polynomial $P$ over a [[Commutative ring|commutative ring]] $A$ is a finite sequence $(a_i)_i$ in $A$. Most of the time, we write it this way
2$$P=\sum_{i=0}^n a_i X^i=a_0+a_1 X+a_2 X^2+\cdots+a_n X^n$$
3where the $a_{i}$ belong to $A$ and $a_{n}\neq 0$.
3where $X$ is called a "formal variable" and $n$ is the biggest integer $k$ such that $a_{k}\neq 0$.
4
5##### Remarks
6- $n$ is called the degree of $P$.
7- The set of all polynomials in $X$ with coefficients in $A$ is denoted $A[X]$.

Revision 802

8/1/2026, 4:50:54 PM · Ancient Tree

Concept edited

This older revision predates detailed metadata tracking.

Revision 798

8/1/2026, 4:35:14 PM · Ancient Tree

Concept edited

This older revision predates detailed metadata tracking.

Revision 794

8/1/2026, 3:33:37 PM · Ancient Tree

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Compare with revision 2953 changed lines
1A polynomial over a [[Commutative ring|commutative ring]] $A$ is a formal expression
2$$P=\sum_{i=0}^n a_i X^i=a_0+a_1 X+a_2 X^2+\cdots+a_n X^n$$
3where the $a_{i}$ belong to $A$ and $a_{n}\neq 0$. $n$ is called the degree of $P$.
3where the $a_{i}$ belong to $A$ and $a_{n}\neq 0$.
4
5##### Remarks
6- $n$ is called the degree of $P$.
6- The set of all polynomials in $X$ with coefficients in $A$ is denoted $A[X]$.

Revision 295

7/7/2026, 2:39:50 PM · Ancient Tree

Concept edited

This older revision predates detailed metadata tracking.

Revision 294

7/7/2026, 2:39:15 PM · Ancient Tree

Concept edited

Compare with revision 1987 changed lines
1A polynomial is...
1A polynomial over a [[Commutative ring|commutative ring]] $A$ is a formal expression
2$$P=\sum_{i=0}^n a_i X^i=a_0+a_1 X+a_2 X^2+\cdots+a_n X^n$$
3where the $a_{i}$ belong to $A$ and $a_{n}\neq 0$. $n$ is called the degree of $P$.
2
3... is called the degree.
5##### Remarks
6- The set of all polynomials in $X$ with coefficients in $A$ is denoted $A[X]$.

Revision 198

7/3/2026, 11:45:32 AM · Ancient Tree

Concept edited

Compare with revision 1962 changed lines
1A polynomial is...
2
3... is called the degree.

Revision 196

7/3/2026, 11:44:55 AM · Ancient Tree

Concept created

A polynomial is...