Factorization of the polynomials

The factorization of a polynomial consists in writing this one like a product of Polynôme S. Ici, polynomial can as well indicate polynomial with a variable or polynomials with several variables.

A polynomial not admitting factorization (with the preceding direction) is known as irreducible ; it is in particular the case of the polynomials of degree 1. A traditional theorem is the existence, on a factorial Anneau, for any nonirreducible polynomial, of a factorization in product of irreducible polynomials; this factorization is primarily single, with permutation close to the factors and invertible near.

Algorithms of factorization of the polynomials with coefficients in the finished bodies are known, for example the Algorithme of Berlekamp.

Irreducible polynomials

Polynomials with coefficients complex

It is a consequence of the Théorème of Alembert-Gauss which mentions that “All Polynôme of degree equal to or higher than 1 to coefficients in the body $\ mathbb C$ of the complex numbers has at least a root in $\ mathbb C$ ”.

Another formulation would consist in saying that the body $\ mathbb {C}$ is algebraically closed.

Consequently, any polynomial of $n$ degree to complex coefficients can be factorized in $\ \ mathbb {C}$ in product of $n$ polynomials of the first degree.

Polynomials with real coefficients

In other words any polynomial $P$ with real coefficients can be factorized in $\ \ R$ in product of polynomials of degree 1 (of which the roots are the real roots of $P$ ) and/or of polynomials of degree 2 without real roots (of which the roots, two to two combined in $\ \ mathbb {C}$ , are the nonreal roots of $P$ ).

The factorization of a polynomial of a variable with real coefficients in irreducible polynomials makes it possible to determine the sign of the values taken by the polynomial function according to the reality in which it is evaluated. This remark is based on the three following properties:

the product of two real nonnull is positive if and only if realities are of the same sign;
the polynomial function of degree 1 $x \ mapsto ax+b$ takes the sign of has for $x>-b/a$ and the sign opposed for $x<-b/a$ ;
the values taken by an irreducible polynomial of degree 2 are constantly sign of the coefficient dominating.

Examples

Let us consider the polynomial $X^4-1 \,$ with coefficients in $\ \ R$ or $\ mathbb {C}$ .

the remarkable identity $a^2-b^2 = (a+b) (a-b) \,$ gives:

X^4-1= (X^2+1) (X^2-1) \,

then:

\ X^4-1= (X^2+1) (X-1) (X+1)

This is factorization in product of irreducible factors to coefficients in

\ \ R

factorization in product of irreducible factors to coefficients in $\ mathbb {C}$ is:

\ X^4-1 = (X+i) (X-i) (X-1) (X+1)

See too

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