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