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Fundamental Theorem of Algebra

The next two result follows directly from the division algorithm .

Bezout’s 1 theorem. If typeset structure is a ring and typeset structure and typeset structure, then the remainder on division of typeset structure by typeset structure is typeset structure.

Note that since we are dividing by a monic polynomial typeset structure, the notes after the Division algorithm for polynomials that we can assume that typeset structure is only a ring.

For an effective version of a calculation of typeset structure the so-called Horner scheme is used..

The next result is actually a corollary of the previous one.

Factor theorem. If typeset structure and typeset structure, then typeset structure is divisible by typeset structure if and only if  typeset structure is a root of typeset structure, symbolically, if and only if typeset structure.

A root typeset structure of a polynomial typeset structure is called

For the sake of simplicity of formulations of some results it is more convenient to consider the simple roots as roots of multiplicity typeset structure.

It can be easily seen that every complex number typeset structure  is a root of a quadratic equation with real coefficients,  namely typeset structure.

The following result follows almost immediately from Factor theorem:

Theorem. A polynomial of degree typeset structure over a field typeset structure has at most typeset structure roots (each counted with multiplicity) in the field typeset structure.

Note that this theorem does not claim, that a given polynomial has at all a root in a given field. The significance of the field typeset structure of complex numbers rest on the fact that the field of complex numbers is the smallest field having the property expressed in the following important result proved by C.F.Gauß  in 1799.

Fundamental theorem of algebra. Any polynomial equation typeset structure with complex coefficient has a complex root.

This result is of a great importance, and cannot be extended to non-polynomial type equations, as the equation typeset structure shows (it has three roots typeset structure).

Note that to be a root is an algebraic property. However, more than 100 proofs of this statement are known and every of them uses a non-algebraic argument, which is more or less dependent on the fact, that a polynomials is a continuous function which is a purely analytical property.This fact is often used as an argument for supporting the unity of all branches of mathematics.

The next result (very often also called as the fundamental theorem of algebra) can be immediately deduced from the previous theorems:

If typeset structure is a non-zero polynomial with complex coefficients of degree typeset structure, then the equation typeset structure has exactly typeset structure complex roots, each counted according to its multiplicity.

This result can also be reformulated in the following way:

If a polynomial typeset structure of degree typeset structure has typeset structureroots then it coincides with zero polynomial, i.e. typeset structure.

Another reformulation says:

If typeset structure and typeset structure are two polynomials over typeset structure such that

Factor Theorem together with Fundamental Theorem of Algebra give:

If typeset structure is a non-zero polynomial with complex coefficients of degree typeset structure, then

f(x) = a _ n(x - α _ 1) (x - α _ 2) ...(x - α _ n)

where typeset structure are all the complex roots of typeset structure, each counted according to its multiplicity.

Notes

1 E. Bezout (1730-1783) French mathematician contributing to the theory of solutions of algebraic equations

1 typeset structure denotes the zero polynomial, i.e. typeset structure.

Cite this web-page as:

Štefan Porubský: Fundamental Theorem of Algebra.

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