General Theory of Algebraic Equations by Etienne Bézout

By Etienne Bézout

This ebook presents the 1st English translation of Bezout's masterpiece, the overall idea of Algebraic Equations. It follows, via nearly 2 hundred years, the English translation of his well-known arithmetic textbooks. the following, Bzout offers his method of fixing platforms of polynomial equations in numerous variables and in nice aspect. He introduces the progressive inspiration of the "polynomial multiplier," which enormously simplifies the matter of variable removal by way of decreasing it to a approach of linear equations. the foremost consequence provided during this paintings, referred to now as "Bzout's theorem," is said as follows: "The measure of the ultimate equation because of an arbitrary variety of whole equations containing an identical variety of unknowns and with arbitrary levels is the same as the fabricated from the exponents of the levels of those equations."The publication deals huge numbers of effects and insights approximately stipulations for polynomials to proportion a typical issue, or to proportion a standard root. It additionally offers a state of the art research of the theories of integration and differentiation of services within the overdue eighteenth century, in addition to one of many first makes use of of determinants to unravel platforms of linear equations. Polynomial multiplier tools became, this day, the most promising techniques to fixing complicated structures of polynomial equations or inequalities, and this translation deals a necessary historical viewpoint in this energetic examine box.

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Mathématiques 1re S et E

Desk des matières :

Chapitre 1. L’outil vectoriel et analytique
    I. Introduction
    II. Le plan vectoriel (rappels)
    III. Les liaisons « plan ponctuel-plan vectoriel »
    IV. L’outil analytique
    V. Compléments
    Exercices

Chapitre 2. L’outil des transformations
    I. Introduction
    II. changes usuelles
    III. motion sur les configurations élémentaires
    IV. modifications associant une determine donnée à une determine donnée
    V. Composition de transformations
    VI. Compléments
    Exercices

Chapitre three. Les angles
    I. Introduction
    II. perspective d’un couple de vecteurs
    III. L’addition des angles
    IV. Propriétés géométriques
    V. Angles et cercles
    VI. Compléments
    Exercices

Chapitre four. Le produit scalaire
    I. Introduction
    II. Produit scalaire de deux vecteurs (rappel)
    III. Produit scalaire en géométrie analytique
    IV. Orthogonalité et cocyclicité
    V. Produit scalaire et lignes de niveau
    VI. Compléments
    Exercices

Chapitre five. Trigonométrie et kinfolk métriques dans le triangle
    I. Introduction
    II. Cosinus et sinus (rappels)
    III. Cosinus et produit scalaire ; sinus et déterminant
    IV. Trigonométrie
    V. relatives métriques dans le triangle
    VI. Compléments
    Trigonométrie (formulaire récapitulatif)
    Exercices

Chapitre 6. Rotations et isométries fixant un element donné
    I. creation (quart de tour)
    II. Rotation de centre O et d’angle α
    III. Rotation : théorèmes de composition et propriétés géométriques
    IV. Isométries fixant un aspect donné
    V. Compléments
    Exercices

Chapitre 7. Le calcul vectoriel dans l’espace
    I. Introduction
    II. L’espace vectoriel E
    III. Droites et plans : repères et vecteurs directeurs
    IV. Éléments de géométrie analytique dans l’espace
    V. Compléments
    Exercices

Chapitre eight. Le produit scalaire dans l’espace
    I. Introduction
    II. Produit scalaire dans E
    III. purposes géométriques du produit scalaire
    IV. Produit scalaire et géométrie analytique
    V. Compléments
    Exercices

Chapitre nine. los angeles sphère
    I. Introduction
    II. los angeles sphère : définition et premières propriétés
    III. part d’une sphère
    IV. Détermination d’une sphère
    V. Surfaces de révolution
    VI. Compléments
    Exercices

Chapitre 10. Statistiques
    I. Introduction
    II. Les caractéristiques de position
    III. Les caractéristiques de dispersion
    IV. Compléments
    Exercices

Extra info for General Theory of Algebraic Equations

Example text

N)T −Q−S − N (u . . n)T −P −Q−S . And among these, the number of those that may also be divided by y R is N (u . . n)T −R−S −N (u . . n)T −P −R−S − N (u . . n)T −Q−R−S +N (u . . n)T −P −Q−R−S . Thus, the number of terms that may be divided by z S but not by uP , nor xQ , nor y R is N (u . . n)T −S − N (u . . n)T −P −S − N (u . . n)T −Q−S + N (u . . n)T −P −Q−S − N (u . . n)T −R−S + N (u . . n)T −P −R−S + N (u . )T −Q−R−S − N (u . . n)T −P −Q−R−S , that is, or, in other terms, dd[N (u .

We also have to substitute the value of x2 y 2 z 2 . 28 BOOK ONE Thus there are usually more possible substitutions to be made in higher order or different incomplete equations. ) We therefore see that the question becomes more complex and that talent and know-how are necessary to persist and use the substitution principle. But we believe we are making a useful remark for analysis by observing that when the quantities whose value is determined through a number of equations have a common divider among them, these values do not summarize all the information provided by these equations.

P, −Q, −R, −S d3 [N (u . . n)T ] . . and so on. ) The expression we have found for the number of terms of interest is not the easiest to handle if we really want to know this number of terms. Should this be the case, we must write this expression explicitly, as we will see in the following example. But, unless I am mistaken, this expression is the most perfect for the purpose we have assigned later. Assume for example that we want to know the number of terms remaining in the polynomial (u . .

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