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In [[algebra]], the '''content''' of a [[polynomial]] with integer coefficients (or, more generally, with coefficients in a [[unique factorization ___domain]]) is the [[greatest common divisor]] of its coefficients. The '''primitive part''' of such a polynomial is the quotient of the polynomial by its content. 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 (ring theory)|unit]] of the [[ring (mathematics)|ring]] of the coefficients (and the multiplication of the primitive part by the [[multiplicative inverse|inverse]] of the unit).
For example, the content of <math>12x^3+30x-20</math> is 2, since 2 is the greatest common divisor of 12, 30, and -20. The primitive part of this polynomial is ▼
:<math>6x^3+15x-10 = \frac{12x^3+30x-20}{2},</math>▼
:<math>12x^3+30x-20 = 2 (6x^3+15x-10).</math>▼
A polynomial is ''[[Primitive polynomial (ring theory)|primitive]]'' if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial.
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Content and primitive part may be generalized to polynomials over the [[rational number]]s, and, more generally, to polynomials over the [[field of fractions]] of a unique factorization ___domain. This makes essentially equivalent the problems of computing greatest common divisors and factorization of polynomials over the integers and of polynomials over the rational numbers.
==Over the integers==
For a polynomial with integer coefficients, the content may be either the [[greatest common divisor]] of the coefficients of its additive inverse. The choice is somehow arbitrary, and may depend on a further convention, which is commonly that the [[leading coefficient]] of the primitive part be positive.
▲For example, the content of <math>-12x^3+30x-20</math>
▲:<math>-6x^3+15x-10 = \frac{-12x^3+30x-20}{2},</math>
and thus the primitive part–content factorization is
▲:<math>-12x^3+30x-20 = 2 (-6x^3+15x-10).</math>
For aesthetically reasons, one often prefer choosing a negative content, here –2, for having the primitive part–content factorization
:<math>-12x^3+30x-20 =-2 (6x^3-15x+10).</math>
==See also==
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