Editing Factorization (section)

===Primitive-part & content factorization===
{{Main|Polynomial factorization#Primitive part–content factorization}}

Every polynomial with [[rational number|rational]] coefficients, may be factorized, in a unique way, as the product of a rational number and a polynomial with integer coefficients, which is [[primitive polynomial (ring theory)|primitive]] (that is, the [[greatest common divisor]] of the coefficients is 1), and has a positive leading coefficient (coefficient of the term of the highest degree). For example:
:<math>-10x^2 + 5x + 5 = (-5)\cdot (2x^2 - x - 1)</math>
:<math>\frac{1}{3}x^5 + \frac{7}{2} x^2 + 2x + 1 = \frac{1}{6} ( 2x^5 + 21x^2 + 12x + 6)</math>

In this factorization, the rational number is called the [[primitive part and content|content]], and the primitive polynomial is the [[primitive part]]. The computation of this factorization may be done as follows: firstly, reduce all coefficients to a common denominator, for getting the quotient by an integer {{mvar|q}} of a polynomial with integer coefficients. Then one divides out the greater common divisor {{mvar|p}} of the coefficients of this polynomial for getting the primitive part, the content being <math>p/q.</math> Finally, if needed, one changes the signs of {{mvar|p}} and all coefficients of the primitive part.

This factorization may produce a result that is larger than the original polynomial (typically when there are many [[coprime integers|coprime]] denominators), but, even when this is the case, the primitive part is generally easier to manipulate for further factorization.