Degree of a polynomial

Definition

Let $R$ be a ring and let $f = a_0 + a_1 x + \dots + a_n x^n$ be a polynomial in $R[x]$. If $f \neq 0$, the degree $\deg f$ of $f$ is the largest integer $d$ with $a_d \neq 0$. The degree of the zero polynomial is $\deg 0 = -\infty$.

Remarks

The degree of $3x^4 - x + 2$ in $\mathbb{Z}[x]$ is $4$. A nonzero constant polynomial has degree $0$. Some authors leave $\deg 0$ undefined, or set it to $-1$. The convention $-\infty$ makes the formula $\deg(f g) = \deg f + \deg g$ hold also for $f = 0$ or $g = 0$. For all $f, g \in R[x]$, $\deg(f + g) \le \max(\deg f, \deg g)$. The coefficient $a_d$ with $d = \deg f$ is the leading coefficient of $f$.

Explanations

L1 human

The degree is the highest power of $x$ that really appears in the polynomial. For example, $x^3 + 5x$ has degree $3$. The zero polynomial has no nonzero coefficient, so it needs a special convention.