Factorization
Factorization
Factorization refers to factoring a polynomial into irreducible polynomials over a given field. Other
factorizations, such as square-free factorization exist, but the irreducible factorization, the most
common, is the subject of this article. It depends strongly on the choice of field. For example, the
fundamental theorem of algebra, which states that all polynomials with complex coefficients have
complex roots, implies that a polynomial with integer coefficients can be completely reduced to linear
factors over the complex field C. On the other hand, such a polynomial may only be reducible to
linear and quadratic factors over the real field R. Over the rational number field Q, it is possible that
no factorization at all may be possible. From a more practical vantage point.
It can be shown that factoring over Q (the rational numbers) can be reduced to factoring over Z (the
integers). This is a specific example of a more general case -- factoring over a field of fractions can
be reduced to factoring over the corresponding integral domain. This algebraic point goes by the name
of Gauss's lemma. The classic proof, due to Gauss, first factors a polynomial into its content, a rational
number, and its primitive part, a polynomial whose coefficients are pure integers and share no
common divisor among them. Any polynomial with rational coefficients can be factored in this way,
using a content composed of the greatest common divisor of the numerators, and the least common
multiple of the denominators. This factorization is unique.
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Obtaining linear factors :- All linear factors with rational coefficients can be found using the rational
root test. If the polynomial to be factored is , then all possible linear factors are of the form , where is
an integer factor of and is an integer factor of . All possible combinations of integer factors can be
tested for validity, and each valid one can be factored out using polynomial long division. If the original
polynomial is the product of factors at least which two of which are of degree 2 or higher, this
technique will only provide a partial factorization; otherwise the factorization will be complete. Note
that in the case of a cubic polynomial, if the cubic is factorable at all the rational root test will give a
complete factorization, either into a linear factor and an irreducible quadratic factor, or into three linear
factors.
Factorizing quartics : -Main article: Quartic function#Factorization into quadratics Reducible quartic
(fourth degree) polynomials with no linear factors can be factored into quadratics.
Duplicate factors : - Main article: Root-finding algorithm#Finding multiple roots of polynomials If
two or more factors of a polynomial are identical to each other, a situation resulting in multiple roots,
then one can exploit the fact that the duplicated factor will also be a factor of the polynomial's
derivative, which itself is a polynomial of one lower degree. The duplicated factor(s) can be found by
using the Euclidean algorithm to find the greatest common factor of the original polynomial and its
derivative.
Kronecker's method :- 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. 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 , and .
Now, 2 can only factor as 1x2, 2x1, (-1)x(-2), or (-2)x(-1).
possible combinations, of which half can be discarded as the negatives of the other half, corresponding
to 64 possible second degree integer polynomials which must be checked. These are the only possible
integer polynomial factors of . Testing them exhaustively reveals. constructed from , and , factors .
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