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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:
:<math>P=c(P)\operatorname{pp}(P).</math>
This shows that every polynomial over the rationals is [[associate elements|associated]] to a unique primitive polynomial over the integers, and that the [[Euclidean algorithm]] allows the computation of this primitive polynomial.
A consequence is that factoring polynomials over the rational is equivalent to factoring primitive polynomials over the integers. As polynomials with coefficients in a [[field (mathematics)}field]] are more common than polynomials with integer coefficient, it may seem that this equivalence may be used for factoring polynomials with integer coefficients. In fact, this is exactly the contrary: every known efficient algorithm for factoring polynomials with rational coefficient uses this equivalence for reducing the problem [[modular arithmetic|modulo]] some prime number {{math|''p''}} (see [[Factorization of polynomials]]).
This equivalence is also used for computing [[polynomial greatest common divisor|greatest common divisors]]s of polynomials, although the [[Euclidean algorithm]] is defined for polynomials with rational coefficients. In fact, in this case, the Euclidean algorithm requires to compute the [[irreducible fraction|reduced form]] of many fractions, and this makes the Euclidean algorithm less efficient than algorithms which work only with polynomials over the integers (see [[polynomial greatest common divisor]]).
==See also==
*[[Rational root theorem]]
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