Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the unit).
Content and primitive part may be generalized to polynomials over the rational numbers, and, more generally, to polynomials over the field of fractions of a unique factorization domain.
The choice is arbitrary, and may depend on a further convention, which is commonly that the leading coefficient of the primitive part be positive.
If one chooses 2 as the content, the primitive part of this polynomial is and thus the primitive-part-content factorization is For aesthetic reasons, one often prefers choosing a negative content, here −2, giving the primitive-part-content factorization In the remainder of this article, we consider polynomials over a unique factorization domain R, which can typically be the ring of integers, or a polynomial ring over a field.
In R, greatest common divisors are well defined, and are unique up to multiplication by a unit of R. The content c(P) of a polynomial P with coefficients in R is the greatest common divisor of its coefficients, and, as such, is defined up to multiplication by a unit.
The primitive part pp(P) of P is the quotient P/c(P) of P by its content; it is a polynomial with coefficients in R, which is unique up to multiplication by a unit.
The primitive-part-content factorization may be extended to polynomials with rational coefficients as follows.
The content of P is the quotient by d of the content of Q, that is and the primitive part of P is the primitive part of Q: It is easy to show that this definition does not depend on the choice of the common denominator, and that the primitive-part-content factorization remains valid: This shows that every polynomial over the rationals is associated with a unique primitive polynomial over the integers, and that the Euclidean algorithm allows the computation of this primitive polynomial.
In fact, the truth is exactly the opposite: every known efficient algorithm for factoring polynomials with rational coefficients uses this equivalence for reducing the problem modulo some prime number p (see Factorization of polynomials).
A polynomial ring over a field is a unique factorization domain.
The same is true for a polynomial ring over a unique factorization domain.
As the content has one less indeterminate, it may be factorized by applying the method recursively.
For factorizing the primitive part, the standard method consists of substituting integers to the indeterminates of the coefficients in a way that does not change the degree in the remaining variable, factorizing the resulting univariate polynomial, and lifting the result to a factorization of the primitive part.