Factoring Polynomials

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The factorization of polynomials in algebra is the decomposition of polynomials into a product of irreducible polynomials , analogous to the prime factorization of whole numbers .

Mathematical description

The aim of factoring is a given polynomial of a polynomial a finite set of irreducible polynomials , to find with


The factors do not all have to be different, that is, the factors can appear in this decomposition with a multiple greater than 1.

If the coefficient ring is a factorial ring , then according to a Gaussian theorem it is also factorial. In this case, there is a system of prime elements , so that this representation is unambiguous except for the sequence and association , and each is an element of the prime system. In rings that are not factorial, it is generally not possible to find a unique factorization.

Over the field of complex numbers , each polynomial -th degree can be described as a product of precisely linear factors

write. This is one of the statements of the fundamental theorem of algebra . It is said that the polynomial breaks down into its linear factors. They are exactly the zeros of the associated polynomial function .

Explanation and examples

Some polynomials can be written as the product of simpler polynomials of smaller degree. For example, factoring out and using a binomial formula results in the decomposition


The factors (occurs twice) and cannot be broken down further: They are irreducible . The polynomial is a divisor of the given polynomial, but it can be broken down even further.

Whether a polynomial is irreducible or can be factored even further depends on the domain of definition of its coefficients under consideration : In the rational numbers, it cannot be further decomposed, in the real numbers it has the factorization . Another example is the polynomial : In the real numbers it is irreducible, in the complex numbers it applies with the imaginary unit .

In general: If a polynomial has a zero , it is divisible by without a remainder , that is, it applies

with a polynomial whose degree is smaller by one and which z. B. can be calculated by polynomial division or with the Horner scheme . If there is now a zero again, this can again be split off as a linear factor. Since a non-constant polynomial always has a zero in the complex numbers according to the fundamental theorem of algebra , this procedure ultimately leads to a factorization by decomposition into linear factors in complex calculations.

Real polynomials

A real polynomial, on the other hand, does not always have a real zero. However, it can be understood as a complex polynomial with real coefficients. As such, it is broken down into linear factors and has the additional property that with every zero the complex conjugate number is also a zero. The two associated linear factors can be converted into the real quadratic polynomial

sum up. This shows that every polynomial in the real numbers can be broken down into a product of linear and quadratic factors. For example, the polynomial has the real zero and the complex conjugate zeros . In the real numbers, its factorization is .

Rational and integer polynomials

For polynomials with integer coefficients, there are various irreducibility criteria , such as the Eisenstein criterion , to determine whether they are irreducible in. The determination of the rational zeros of a polynomial

can be solved algorithmically in a finite number of steps, because for every zero it holds that it is a divisor of and a divisor of (see theorem about rational zeros ).

For example, in the case of the polynomial, you can find the rational zero by trying out all the possibilities . Polynomial division results

and the polynomial is irreducible according to the Eisenstein criterion (with the prime number 2), so that ultimately the integer factorization



BA Hausmann described an application of Kronecker's algorithm in 1937 . Elwyn Berlekamp published the Berlekamp algorithm in 1967 , with which polynomials can be factored over the remainder class field . In 1992, Harald Niederreiter discovered another way of factoring polynomials over finite fields; the Niederreiter algorithm can be traced back to him .

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