The Vieta theorem and a bit of history

The theorem of Vieta - this concept is familiar from school times to virtually everyone. But is it "really" familiar? Few people face it in everyday life. But not all those who deal with mathematics, sometimes fully understand the profound meaning and great importance of this theorem.

The Vieta theorem in many ways facilitates the process of solving a huge number of mathematical problems, which eventually reduce to the solution of the quadratic equation :

Ax2 + bx + c = 0 , where a ≠ 0.

This is the standard form of the quadratic equation. In most cases, the quadratic equation has coefficients a , b , and c , which can be easily simplified by dividing them by a . In this case, we come to the form of a quadratic equation, called reduced (when the first coefficient of the equation is 1):

X2 + px + q = 0

It is for this type of equations that the Vieta theorem is convenient to use. The main sense of the theorem is that the values of the roots of the reduced square equation can be easily defined orally, knowing the basic relationship of the theorem:

• The sum of the roots is equal to the number opposite to the second coefficient (ie -p);
• The product is equal to the third coefficient (i.e., q).

Namely, x1 + x2 = -p and x1 * x2 = q .

The solution of most problems in the school course of mathematics is reduced to simple pairs of numbers that are easily found in the possession of minimal oral computing skills. And this should not cause any problems. The existing inverse theorem of Viete makes it possible to reconstruct its coefficients and the record in the standard form from the available pair of numbers that are the roots of some quadratic equation.

The ability to use Viet's theorem as an instrument greatly facilitates the solution of mathematical and physical problems in the secondary school course. Especially this skill is indispensable in the preparation of high school students for the USE.

Realizing the importance of such a simple and effective mathematical tool, you involuntarily think about the person who first opened it.

François Viet is a famous French scientist who started his career as a lawyer. But, obviously, mathematics was his vocation. While in the royal service as an adviser, he was famous for having managed to read the intercepted cryptic message of the King of Spain to the Netherlands. This gave the French King Henry III the opportunity to know all the intentions of his opponents.

Gradually becoming familiar with mathematical knowledge, Francois Viete came to the conclusion that there must be a close connection between the newest at the time the research of "algebraists" and the deep geometric legacy of the ancients. In the course of scientific research he developed and formulated almost all elementary algebra. He first introduced the use of letter magnitudes in a mathematical apparatus, clearly delineating concepts: number, magnitude and their relationship. Viet has proved that, performing operations in symbolic form, it is possible to solve the problem for the general case, practically for any values of given values.

His research for the solution of equations of higher degree than the second, resulted in the theorem, which is now known as the generalized theorem of Vieta. It has great practical significance, and its application makes it possible to quickly solve equations of higher order.

One of the properties of this theorem is the following: the product of all the roots of the equation of the nth power is equal to its free term. This property is often used in solving equations of the third or fourth degree in order to reduce the order of the polynomial. If the nth degree polynomial has integer roots, then they can be easily determined by the simple selection method. And then, after dividing the polynomial by the expression (x-x1), we get a polynomial (n-1) -th power.

In the end, it should be noted that Vieta's theorem is one of the most famous theorems of the school course of algebra. And his name takes a worthy place among the names of great mathematicians.