# Definition:Leading Coefficient of Polynomial

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## Definition

Let $R$ be a commutative ring with unity.

Let $P \in R \sqbrk X$ be a nonzero polynomial over $R$.

Let $n$ be the degree of $P$.

The **leading coefficient** of $P$ is the coefficient of $x^n$ in $P$.

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

Let $\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ be a polynomial in $x$ over $S$.

The coefficient $a_n \ne 0_R$ is called the **leading coefficient** of $f$.

### Polynomial Form

Let $R$ be a commutative ring with unity.

Let $f = a_0 + a_1 X + \cdots + a_{r-1} X^{r-1} + a_r X^r$ be a polynomial form in the single indeterminate $X$ over $R$.

Then the ring element $a_r$ is called the **leading coefficient** of $f$.

## Also see

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 64$. Polynomial rings over an integral domain