Definition:Coefficient
Definition
A coefficient is a constant which is used in a particular context to be multiplied by a variable that is under consideration.
Binomial Coefficient
Let $n \in \Z_{\ge 0}$ and $k \in \Z$.
Then the binomial coefficient $\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\paren {a + b}^n$.
Polynomial Coefficient
For the usage of this term in the context of polynomial theory:
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 $\set {x^k : k \in \N}$.
Also see
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): coefficient