Definition:Coefficient of Polynomial

From ProofWiki
Jump to navigation Jump to search

One variable

General definition

Let $R$ be a commutative ring with unity.

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

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

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

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 $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

Let $x \in R$.

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

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

Also see