# Factoring Polynomials

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

results.

## Algorithms

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 .