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.

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 $\paren {a + b}^n$.

For the usage of this term in the context of polynomial theory:

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 $R \sqbrk x$.

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 $\set {x^k : k \in \N}$.

The terms:

$a_0, a_1, \ldots, a_n, \ldots$

are the coefficients of $\ds \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$.

A coefficient, in the context of physics, is a number that is used as a measure of some property of a body.