Vieta's theorem - this concept is familiar from school times to almost everyone. But is it really “familiar”? Few people encounter it in everyday life. But not all those who deal with mathematics sometimes fully understand the deep meaning and enormous significance of this theorem.
Vieta's theorem greatly facilitates the process of solving a huge number of mathematical problems, which ultimately come down to solving 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 that 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 Vieta's theorem is convenient to use. The main meaning of the theorem is that the values of the roots of the reduced quadratic equation can be easily determined verbally, knowing the main relation 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 to most problems in the school course of mathematics comes down to simple pairs of numbers that are easily found with minimal oral computing skills. And this should not cause any problems. The existing converse Vieta theorem allows you to easily restore its coefficients and writing in standard form from the existing pair of numbers that are the roots of some quadratic equation.
The ability to use the Vieta theorem as a tool greatly facilitates the solution of mathematical and physical problems in a high school course. Especially this skill is indispensable in preparing high school students for the exam.
Having understood the significance of such a simple and effective mathematical tool, you involuntarily think about the person who first discovered it.
Francois Viet is a famous French scientist who began his career as a lawyer. But, obviously, mathematics was his calling. While in the royal service as an adviser, he became famous for being able to read the intercepted encrypted message of the King of Spain to the Netherlands. This gave the French king Henry III the opportunity to know about all the intentions of his opponents.
Gradually familiarizing himself with mathematical knowledge, François Viet came to the conclusion that there should be a close relationship between the latest research of the "algebraists" at that time and the deep geometric heritage of the ancients. In the course of scientific research, he developed and formulated almost all elementary algebra. He first introduced the use of alphabetic values in the mathematical apparatus, clearly distinguishing between concepts: number, magnitude, and their relationship. Viet proved that, performing operations in a symbolic form, it is possible to solve the problem for the general case, for almost any value of a given value.
His research for solving equations of greater degrees than the second resulted in a theorem, which is now known as the generalized Vieta theorem. It is of great applied value, and its application makes it possible to quickly solve equations of a higher order.
One of the properties of this theorem is the following: the product of all the roots of an equation of the nth degree is equal to its free term. This property is often used in solving equations of the third or fourth degree in order to lower the order of the polynomial. If an nth degree polynomial has integer roots, then they can be easily determined by simple selection. And then, after dividing the polynomial by the expression (x-x1), we obtain a polynomial of the (n-1) -th degree.
In the end, I want to note that the Vieta theorem is one of the most famous theorems of the school course of algebra. And his name occupies a worthy place among the names of the great mathematicians.