Francois Viete, a French mathematician, established the relationship between the roots and coefficients of an equation in his book On the Identification and Correction of Equations, and put forward this theorem. [2] Because David first discovered this relationship between the roots and coefficients of algebraic equations, people call this relationship the Vieta Theorem. [3]
Chinese name
Vieta theorem
Foreign name
Vieta theorem
presenter
Francois Viete
Show time
16 15
Applied discipline
Mathematical algebra
quick
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Mathematical deduction
Theorem generalization
A brief history of development
Theorem significance
Theorem relation
We assume that in the unary quadratic equation, two x 1 and x2 have the following relationship:
[4]
Mathematical deduction
From the root formula of quadratic equation in one variable:
Examples of application of Vieta theorem
There are:
[4]
Theorem generalization
Inverse principle
If two numbers α and β satisfy the following relationship: α+β =, α β =, then these two numbers α and β are the roots of the equation.
Through the inverse theorem of Vieta's theorem, we can use the sum-product relation of two numbers to construct a quadratic equation with one variable. [5]
Generalization theorem
Vieta's theorem can not only explain the relationship between roots and coefficients of a quadratic equation with one variable, but also generalize the relationship between roots and coefficients of a quadratic equation with one variable.
Theorem:
Let the root of a univariate equation with n complex coefficients be, then it holds:
That is, the sum of all roots is the reciprocal of the ratio of (n- 1) degree coefficient to n degree coefficient, and the product of all roots is the ratio of constant term to n degree coefficient multiplied by (-1) n.
Note: the proof of this generalized form cannot be carried out according to the root formula, because there is no root formula for univariate equations with more than five degrees. The proof steps are complicated. By decomposing the polynomial factor on the left, removing the brackets and comparing the coefficients of the same term, a conclusion is drawn. This method is universal. Even if an equation has a root formula, Vieta's theorem can be proved by this method without the help of the root formula. [6]
A brief history of development
Francois Viete, a French mathematician, improved the solutions of cubic and quartic equations in his book On the Identification and Revision of Equations, and established the relationship between the roots and coefficients of equations when n=2 and 3, which is the modern Vieta theorem. [2]
Francois Viete
David first discovered this relationship between the roots and coefficients of algebraic equations, so people called this relationship Vieta Theorem. David got this theorem in16th century. The proof of this theorem depends on the basic theorem of algebra, but the basic theorem of algebra was first discussed by Gauss in 1799. [3]
Theorem significance
Vieta's theorem plays a unique role in finding the symmetric function of roots, discussing the sign of roots of quadratic equations, solving symmetric equations and solving some conic problems.
The discriminant of the root of a quadratic equation with one variable is (A, B and C are quadratic coefficient, linear coefficient and constant term of the quadratic equation with one variable respectively). The relationship between Vieta's theorem and the discriminant of roots is even more inseparable.
The discriminant of roots is a necessary and sufficient condition for judging whether an equation has real roots. Vieta theorem explains the relationship between roots and coefficients. Whether the equation has real roots or not, Vieta's theorem is applicable between the roots and coefficients of a quadratic equation with real coefficients. The combination of discriminant and Vieta's theorem can more effectively explain and judge the conditions and characteristics of the roots of a quadratic equation with one variable. [ 1]
The most important contribution of Vieta's theorem is the promotion of algebra. Firstly, he systematically introduced algebraic symbols, promoted the development of equation theory, replaced unknowns with letters, and pointed out the relationship between roots and coefficients. Vieta's theorem laid a foundation for the study of the unary equation in mathematics, and created and opened up a broad development space for the application of the unary equation. [7]
Vieta's theorem can be used to quickly find the relationship between the roots of two equations. Vieta's theorem is widely used in elementary mathematics, analytic geometry, plane geometry and equation theory. [4] [8] [9]
reference data
[1] Hu Peiming. On the joint application of the discriminant of quadratic equation roots and Vieta's theorem [J]. Curriculum Education Research, 20 14 (4): 134.
[2] History of foreign mathematics. Middle School Subject Network.2008-11-03 [reference date 20 14-08-27]
[3] Zhao Shihong. On the application of Vieta theorem [J]. Journal of Taiyuan City Vocational and Technical College, 2008 (7): 100.