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
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Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {S, +, \circ}$ be a subring of $R$.
Let $x \in R$.
Let $\ds f = \sum_{j \mathop = 0}^n \paren {a_j \circ x^j} = a_0 + a_1 \circ x + a_2 \circ x^2 + \cdots + a_{n - 1} \circ x^{n - 1} + a_n \circ x^n$ be a polynomial in $x$ over $R$.
The elements of the set $\set {a_0, a_1, \ldots, a_n}$ are the coefficients of $f$.
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polynomial
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polynomial