---
tag: 0P05RJC5XA
kind: definition
name: Leading coefficient
msc: [13F20]
defines: 0V5GYWR8R8
uses: [0DPF6WRZW6]
sources:
  - url: https://en.wikipedia.org/wiki/Coefficient
    locator: Terminology and definition
---

## Definition

Let $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a nonzero polynomial in $R[x]$ of [[degree|0DPF6WRZW6]] $d$.
The leading coefficient of $f$ is $a_d$.
The leading coefficient of the zero polynomial is $0$.

## Remarks

The leading coefficient of $3x^4 - x + 2$ is $3$.
A polynomial with leading coefficient $1$ is [[monic|0BDX4MBKT4]].
In an [[integral domain|09S3PFBEPM]], the leading coefficient of $f g$ is the product of the leading coefficients of $f$ and $g$.

## Explanations

### L1 human

The leading coefficient is the number in front of the highest power of $x$.
For large values of $x$, the leading term $a_d x^d$ decides the size of a real polynomial.