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==Properties==
In the remaining of this article, we consider polynomials over a [[unique factorization ___domain]] {{math|''R''}}, which can typically be the ring of [[integer]]s, or a [[polynomial ring]] over a [[field (mathematics)|field]]. In {{math|''R''}}, [[greatest common divisor]] are well defined, and are unique [[up to]] the multiplication by a [[unit (ring theory)|unit]] of {{math|''R''}}.
The '''content''' {{math|''c''(''P'')}} of a polynomial {{math|''P''}} with coefficients in {{math|''R''}} is the greatest common divisor of its coefficients, and, as such, is defined up to the multiplication by a unit. The '''primitive part''' {{math|pp(''P'')}} of {{math|''P''}} is the quotient {{math|''P''/''c''(''P'')}} of {{math|''P''}} by its content; it is a polynomial with coefficients in {{math|''R''}}, which is unique up to the multiplication by a unit. If the content is changed by multiplication by a unit {{math|''u''}}, then the primitive part must be changed by dividing it by the same unit, in order to keep the equality
:<math>P=c(P)\operatorname{pp}(P),</math>
which is called the prime-part-content factorization of {{math|''P''}}.
The main properties of the content and the primitive part result of [[Gauss's lemma (polynomial)|Gauss's lemma]], which asserts that the product of two primitive polynomials is primitive, where a polynomial is primitive if 1 is a greatest common divisor of its coefficients. This implies:
*The content of a product of polynomial is the product of their contents:
::<math>c(P_1P_2)=c(P_1)c(P_2)</math>
*The primitive part of a product of polynomials is the product of their primitive parts:
::<math>\operatorname{pp}(P_1P_2)=\operatorname{pp}(P_1)\operatorname{pp}(P_2)</math>
*The content of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their contents:
::<math>c(\operatorname{gcd}(P_1, P_2))=\operatorname{gcd}(c(P_1), c(P_2))</math>
*The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in {{math|''R''}}) of their primitive parts:
::<math>\operatorname{pp}(\operatorname{gcd}(P_1, P_2))=\operatorname{gcd}(\operatorname{pp}(P_1), \operatorname{pp}(P_2))</math>
*The complete [[factorization of polynomials|factorization]] of a polynomial over {{math|''R''}} is the product of the factorization (in {{math|''R''}}) of the content and of the factorization (in the polynomial ring) of the primitive part.
The last property imply that the computation of the prime-part-content factorization of a polynomial reduces the computation of its complete factorization to the separate factorization of the content and the primitive part. This is generally interesting, because the computation of the prime-part-content factorization involves only greatest common divisor computation in {{math|''R''}}, which is usually much easier than factorization.
==See also==
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