SOLVING EQUATIONS OF THE THIRD DEGREE

​ABSTRACT

Importance Many great scientists and mathematicians have studied equations of the third degree. Among them are Evariste Galois, Karl Friedrich Gauss, Leonhard Euler, Alexander Grothendieck, Alan Turing, Nikolai Ivanovich Lobachevsky, Girardo Gerolamo Cardano and many others. Each of these scientists has made an invaluable contribution to the development of this field, revealing in his writings the essence of equations and suggesting ways to solve them. This article will discuss some methods for solving equations of the third degree, and also propose tasks for monitoring knowledge on this topic.

Objectives Comprehensive study of methods for solving cubic equations. Development of tasks for knowledge monitoring.

Methods In the process of studying methods for solving equations of the third degree, theoretical, observational methods.

Results Methods for solving both complete and incomplete cubic equations were considered. An equation always has at least one root; the other two roots will either be real, or a complex conjugate pair, or the equation will have no other roots. A set of tasks was developed in the form of a test, including theoretical and practical parts, aimed at monitoring knowledge and skills on the topic "Solving equations of the third degree".

Keywords: equations of the third degree, solution methods, knowledge monitoring.

In the school course of mathematics (algebra), attention is paid to various types of equations, but, coming to the ninth grade, children get acquainted with such type of equations as equations of the third degree or cubic equations. Further study of this topic takes place within the framework of algebra courses of grades 10-11, as well as courses of institutes of mathematical orientation.

Third degree equation (or cubic equation): "...refers to equations of the nth degree, which mean equations of the form p(x) = 0, where p(x) = 0, где p(x) = a₀xⁿ + a₁x^n-1 + … + a_n-1x + a_n – a polynomial of degree n, where a₁ a₂ ... a_n – are the given real numbers, and a ≠ 0"[1]. The reduced cubic equation is the equation where the coefficient a = 1, it has the form: x³+bx²+cx+d=0.

The root of the cubic equation is the number x, when substituting it into the equation, it turns into a true equality. A cubic equation always has at least one real root; the other two roots will either be real, or a complex conjugate pair, or will be absent.

Binomial cubic equation: "This is an equation of the form ax³ + b = 0"[1]. Its solution is based on bringing the equation into the form  by dividing by a coefficient an other than zero. Next, we apply the reduced multiplication formula "sum of cubes", obtaining this equality: . From the first bracket we learn that , and the quadratic equation has no roots in the domain of real numbers, since its discriminant . Therefore, the equation has only one root, which is .

Consider the following method. It is worth noting in advance that the graphical method of solving cubic equations has a number of significant drawbacks, which include the high complexity of plotting some equations and, in some cases, the low accuracy of the answer.

This method is based on the construction of graphs of functions and the search for their intersection points, which will be the roots of our equation. For example, to find the roots of the equation , we will write this equation in the form, and then build two graphs in one coordinate system:  and . The intersection points of these graphs will be the desired roots.

The returnable cubic equation is "an equation of the form ax3+bx2+cx+d=0, in which the coefficients standing on the symmetric relative midpoints are equal, that is, a = d, and b = c"[1]. In such equations, the left part is decomposed into multipliers: . It should be remembered that such an equation will always have a root x = -1, and the remaining roots are found according to the algorithm for finding the roots of a quadratic equation.

The method of trigonometric substitution for solving equations of the third degree is used with if the domain of definition of the equation being solved is included in the domain of definition of the trigonometric function or coincides with it, that is, the roots of our equation are possible only from the interval (-1;1), and this means that we can introduce a replacement x=for t∈(0;π).

Based on this material, tasks have been developed that can be used during math lessons, extracurricular activities, as well as for self-testing and monitoring of one's own knowledge.

Tasks for knowledge monitoring:

1. The reduced cubic equation is the equation where the coefficient a =

a) 3

b) 2

c) 6

d) 1

2. The root of a binomial cubic equation is calculated by the formula:

a)

b)

c)

d)

3. The roots of the equation of the third degree when solving it by the graphical method will be the points:

a) Intersections of graphs of functions with coordinate axes.

b) Intersections of graphs of functions that make up the equation of the third degree.

c) Intersections of the quadratic equation with the coordinate axes.

d) On which the function decreases.

4. The method of trigonometric substitution in solving equations of the third degree can be used provided that:

1. The domain of definition of the equation to be solved is not included in the domain of definition of the trigonometric function and does not coincide with it.
2. The domain of definition of the equation to be solved is included in the domain of definition of the trigonometric function, but does not coincide with it.
3. The domain of definition of the equation to be solved is included in the domain of definition of the trigonometric function or coincides with it.
4. If the domain of definition of the equation to be solved is the interval (2;3).

5. The returnable equation of the third degree is the equation:

a)

b)

c)

d)

The practical part:

1. Solve the binomial cubic equation .

When solving this equation, we must recall the formula of the roots of the binomial equation (this question was in the theoretical part of the tasks). With it, we can immediately calculate a single root of this equation: . But for reliability, let's double-check whether this is the case by decomposing the equation into the product of a quadratic trinomial and a binomial, as was done when considering a binomial cubic equation. To reduce the number of calculations, we immediately write  as 4, thereby obtaining the product . Solving the quadratic equation through the discriminant, we make sure that it has no roots, because – which means no roots.

The answer to task 1: -4.

2. Solve the cubic equation .

As we can see, this equation refers to the returnable (or symmetric) equations. This type of equation is decomposed into the product of a binomial and a square trinomial by the formula . Let 's carry out the decomposition and solve the equation: .

As we can already see, the equation has a root x1 = -1. Now consider the quadratic equation and calculate its discriminant:  thus, we are convinced that the quadratic equation has no roots, and, as a consequence, the cubic equation has alone root = 1.

Conclusion. Methods for solving both complete and incomplete cubic equations were considered. An equation always has at least one root; the other two roots will either be real, or a complex conjugate pair, or the equation will have no other roots. A set of tasks was developed in the form of a test, including theoretical and practical parts, aimed at monitoring knowledge and skills on the topic "Solving equations of the third degree".

References:

1. Mathematical Encyclopedic Dictionary (1988) / Editor-in-chief Yu. V. Prokhorov; executive editor S.I. Adyan, N.S. Bakhvalov, V.I. Biryukov, A.P. Ershov, L.D. Kudryavtsev, A.L. Onishchik, A.P. Yushkevich.