# Definition:Coefficient of Polynomial

## One variable

### General definition

Let $R$ be a commutative ring with unity.

Let $P \in R \sqbrk x$ be a polynomial over $R$.

By Monomials Form Basis of Polynomial Ring, the set $\set {x^k : k \in \N}$ is a basis of $R \sqbrk x$.

By Equality of Monomials of Polynomial Ring, all $x^k$ are distinct.

The coefficient of $x^k$ in $P$, or the $k$th coefficient of $P$, is the $x^k$-coordinate of $P$ with respect to the basis $\set {x^k : k \in \N}$.

### Polynomial Form

Let $R$ be a commutative ring with unity.

Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

The ring elements $a_1, \ldots, a_r$ are the coefficients of $f$.

## Polynomial in Ring Element

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

Let $x \in R$.

Let $\displaystyle f = \sum_{j \mathop = 0}^n \paren {a_j \circ x^j} = a_0 + a_1 \circ x + a_2 \circ x^2 + \cdots + a_{n - 1} \circ x^{n - 1} + a_n \circ x^n$ be a polynomial in $x$ over $R$.

The elements of the set $\set {a_0, a_1, \ldots, a_n}$ are the coefficients of $f$.