Content of a polynomial

Definition

Let $R$ be a unique factorisation domain and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$. The content of $f$, written $c(f)$, is a greatest common divisor of the coefficients $a_0, a_1, \dots, a_n$.

Remarks

The content is unique up to associates. Over $\mathbb{Z}$, one often takes the positive gcd, with the sign of the leading coefficient in the primitive part. For $f = 6x^2 + 4x - 2$ in $\mathbb{Z}[x]$, $c(f) = 2$. Each nonzero $f$ is $c(f) \cdot g$ with $g$ a primitive polynomial, the primitive part of $f$. The content is multiplicative up to units: $c(f g) \sim c(f) c(g)$. This is a form of Gauss's lemma.

Explanations

L1 human

The content is the largest common factor that you can pull out of all the coefficients. For $6x^2 + 4x - 2$, you can pull out $2$ and get $2(3x^2 + 2x - 1)$.