Factorization of polynomials: Difference between revisions

Revision as of 22:02, 19 January 2006

Revision as of 23:20, 19 January 2006

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Now, any polynomial with rational coefficients can be split into a content and a primitive polynomial, and in particular the factors of any factorization (over '''Q''') of such a polynomial can also be so split. Since the content and the primitive polynomials are unique, and since the product of primitive polynomials is itself primitive, the primitive part of the polynomial must factor into the primitive parts of the factors. In particular, if a polynomial with integer coefficients can be factored at all, it can be factored into integer polynomials (proof). So factoring a polynomial with rational coefficients can be reduced to finding integer factorizations of its primitive part over '''Z''' and finding prime factorizations of the integers in its content.

Currently the best techniques for factoring integer polynomials involve factoring over finite fields, but a simpler technique is usable for small polynomials (roughly less than tenth degree), if a computer is used. Since integer polynomials must factor into integer polynomial factors, and evaluating integer polynomials at integer values must produce integers, the integer values of a polynomial can be factored in only a finite number of ways, and produce only a finite number of possible polynomial factors.

For example, consider <math>f(x) = x^5 + x^4 + x^2 + x + 2</math>. If this polynomial factors over '''Z''', then at least one of its factors must be of degree two or less. We need three values to uniquely fit a second degree polynomial. We'll use <math>f(0) = 2</math>, <math>f(1) = 6</math> and <math>f(-1) = 2</math>. Now, 2 can factor as either 1×2, 2×1, -1×-2, or -2×-1. Therefore, if a second degree integer polynomial factor exists, it must take one of the values 1, 2, -1, or -2 at <math>x=0</math>, and likewise at <math>x=-1</math>. There are eight different ways to factor 6, so there are 4×4×8 = 64 possible combinations,

of which half can be discarded as the negatives of the other half, corresponding to 32 possible second degree integer polynomials which must be checked. These are the only possible integer polynomial factors of <math>f(x)</math>. Testing them exhaustively reveals that <math>g(x) = x^2+x+1</math>, constructed from <math>g(0)=1</math>, <math>g(1)=2</math> and <math>g(-1)=1</math>, factors <math>f(x)</math>.

(''Van der Waerden'', Sections 5.4 and 5.6)