Vieta's formulas

The set of Vieta or root set of Vieta is a mathematical theorem of elementary algebra. It is named after the mathematician François Viète who proved it in his book " De aequationum recognitione et emendatione Tractatus duo". The set makes a statement about the relationship of the coefficients of an algebraic equation and its solutions.

Statement

Are the coefficients of the quadratic equation and

And and their solutions ( roots). Then we have

Examples

For the record, there are three important applications:

  • It can thus construct quadratic equations at predetermined solutions. For example, is a quadratic equation for the solutions 2 and 3.
  • There can be systems of equations of the form
  • The set can help to determine the solutions by trial and error: If the quadratic equation

Evidence

The rate is calculated directly by multiplying the zeros form by comparing coefficients:

And thus and.

Generalization

The set of Vieta about quadratic equations can be generalized to polynomial equations and polynomials of arbitrary degree. This generalization of the theorem of Vieta is the basis for solving equations of higher degree by polynomial division. By the fundamental theorem of algebra applies:

Every (normalized ) polynomial of degree n with coefficients in the complex numbers can be represented as a product of n linear factors:

X1, x2, ..., xn are the zeros of the polynomial; even if all the coefficients a0, a1, ... are real, the roots can be complex. Not all xi must be different.

Well, there is the set of Vieta by multiplying out and equating coefficients:

In which

The so-called elementary symmetric polynomials in up are. For a fourth-degree polynomial

Results in:

An important application of the set of n = 2 and n = 3 in the return of the cubic equation of a quadratic equation, and the equation of the fourth degree of a cubic equation, the so-called cubic resolvent.

Generally, the root set of Vieta also applies to polynomials with coefficients in other bodies, as long as they are completed only algebraically.

710312
de