Factorization of polynomials
As a factorization of polynomials in algebra is understood analogously to the prime factorization of integers disassembly of polynomials into a product of irreducible polynomials.
The aim of the factorization is to find for a given polynomial of a polynomial is a finite set of irreducible polynomials with
The factors need not be all different while, that is, the factors may arise with a multiplicity greater than 1 in this decomposition.
If the coefficient ring is a factorial ring, then is also factorial by a set of Gaussian. In this case, there exists a system of prime elements, so this representation up to the order and Associated awareness is unique and each is an element of Primsystems. In rings that are not factorial, it is generally not possible to find a unique factorization.
Over the field of complex numbers can be any polynomial of degree n as a product of linear factors exactly
. Write This is one of the statements of the Fundamental Theorem of Algebra. They say that the polynomial splits into its linear factors. Those are exactly the zeros of the corresponding polynomial function.
Explanation and examples
Some polynomials can be written as a product of simpler polynomials of lower degree. For example, is obtained by factoring out and using a binomial formula, the decomposition
The factors (occurs twice on ), and can not be split further: they are irreducible. The polynomial is indeed a divisor of the given polynomial, but it can be even further decompose.
Whether or not a polynomial is irreducible or can be factored further, depends on the considered domain of definition of its coefficients: Taking the example of the rational numbers not split further, into the real numbers it has the factorization. Another example is the polynomial: In the real numbers, it is irreducible, however applies to the complex numbers with the imaginary unit.
As a general rule: If a polynomial is a zero, then it is divisible without remainder, that is, it is
With a polynomial whose degree is smaller by one and that may be calculated, for example by polynomial or the Horner scheme. Now has a zero again, then can in turn split off as a linear factor. As in the complex numbers according to the fundamental theorem of algebra a non-constant polynomial always has a zero, this approach eventually leads to complex invoice factoring by decomposition into linear factors.
A real polynomial does not always has a real root against. However, it can be a complex polynomial with real coefficients understand. As such, it is divided into linear factors and additionally has the property that with each zero and the complex conjugate number is a zero. The two associated linear factors can be to the real quadratic polynomial
Summarized. Thus we have shown that every polynomial can be decomposed into a product of linear and quadratic factors in the real numbers. For example, the polynomial has the real root and the complex conjugate zeros. In the real numbers is its factorization.
Rational and integer polynomials
For polynomials with integer coefficients are different Irreduzibilitätskriterien, such as the Eisenstein criterion to determine whether they are irreducible. The determination of the rational zeros of a polynomial
Can be algorithmically solved in a finite number of steps, as is true for every zero that divides and divides is.
For example, one finds the rational zero at the polynomial by trying all possibilities. polynomial yields
And the polynomial is after the Eisenstein criterion ( with the prime 2) is irreducible, so that finally the integer factorization
Elwyn Berlekamp Ralph published in 1967 the Berlekamp algorithm, can be factored over the residue field with the polynomials.
BA Hausmann described in 1937 an application of the algorithm of Kronecker.