Solving equations of higher degrees using various methods. Equations of higher degrees Methods for solving equations n

Let's consider solving equations with one variable of degree higher than the second.

The degree of the equation P(x) = 0 is the degree of the polynomial P(x), i.e. the greatest of the powers of its terms with a coefficient not equal to zero.

So, for example, the equation (x 3 – 1) 2 + x 5 = x 6 – 2 has the fifth degree, because after the operations of opening the brackets and bringing similar ones, we obtain the equivalent equation x 5 – 2x 3 + 3 = 0 of the fifth degree.

Let us recall the rules that will be needed to solve equations of degree higher than two.

Statements about the roots of a polynomial and its divisors:

1. Polynomial nth degree has a number of roots not exceeding n, and roots of multiplicity m occur exactly m times.

2. A polynomial of odd degree has at least one real root.

3. If α is the root of P(x), then P n (x) = (x – α) · Q n – 1 (x), where Q n – 1 (x) is a polynomial of degree (n – 1).

4.

5. The reduced polynomial with integer coefficients cannot have fractional rational roots.

6. For a third degree polynomial

P 3 (x) = ax 3 + bx 2 + cx + d one of two things is possible: either it is decomposed into the product of three binomials

Р 3 (x) = а(х – α)(х – β)(х – γ), or decomposes into the product of a binomial and a square trinomial Р 3 (x) = а(х – α)(х 2 + βх + γ ).

7. Any polynomial of the fourth degree can be expanded into the product of two square trinomials.

8. A polynomial f(x) is divisible by a polynomial g(x) without a remainder if there is a polynomial q(x) such that f(x) = g(x) · q(x). To divide polynomials, the “corner division” rule is used.

9. For the polynomial P(x) to be divisible by a binomial (x – c), it is necessary and sufficient that the number c be the root of P(x) (Corollary of Bezout’s theorem).

10. Vieta's theorem: If x 1, x 2, ..., x n are real roots of the polynomial

P(x) = a 0 x n + a 1 x n - 1 + ... + a n, then the following equalities hold:

x 1 + x 2 + … + x n = -a 1 /a 0,

x 1 x 2 + x 1 x 3 + … + x n – 1 x n = a 2 /a 0,

x 1 x 2 x 3 + … + x n – 2 x n – 1 x n = -a 3 / a 0,

x 1 · x 2 · x 3 · x n = (-1) n a n / a 0 .

Solving Examples

Example 1.

Find the remainder of division P(x) = x 3 + 2/3 x 2 – 1/9 by (x – 1/3).

Solution.

By corollary to Bezout’s theorem: “The remainder of a polynomial divided by a binomial (x – c) is equal to the value of the polynomial of c.” Let's find P(1/3) = 0. Therefore, the remainder is 0 and the number 1/3 is the root of the polynomial.

Answer: R = 0.

Example 2.

Divide with a “corner” 2x 3 + 3x 2 – 2x + 3 by (x + 2). Find the remainder and incomplete quotient.

Solution:

2x 3 + 3x 2 – 2x + 3| x + 2

2x 3 + 4 x 2 2x 2 – x

X 2 – 2 x

Answer: R = 3; quotient: 2x 2 – x.

Basic methods for solving equations higher degrees

1. Introduction of a new variable

The method of introducing a new variable is already familiar from the example of bi quadratic equations. It consists in the fact that to solve the equation f(x) = 0, a new variable (substitution) t = x n or t = g(x) is introduced and f(x) is expressed through t, obtaining a new equation r(t). Then solving the equation r(t), the roots are found:

(t 1, t 2, …, t n). After this, a set of n equations q(x) = t 1 , q(x) = t 2 , … , q(x) = t n is obtained, from which the roots of the original equation are found.

Example 1.

(x 2 + x + 1) 2 – 3x 2 – 3x – 1 = 0.

Solution:

(x 2 + x + 1) 2 – 3(x 2 + x) – 1 = 0.

(x 2 + x + 1) 2 – 3(x 2 + x + 1) + 3 – 1 = 0.

Substitution (x 2 + x + 1) = t.

t 2 – 3t + 2 = 0.

t 1 = 2, t 2 = 1. Reverse substitution:

x 2 + x + 1 = 2 or x 2 + x + 1 = 1;

x 2 + x - 1 = 0 or x 2 + x = 0;

Answer: From the first equation: x 1, 2 = (-1 ± √5)/2, from the second: 0 and -1.

2. Factorization by grouping and abbreviated multiplication formulas

The basis this method is also not new and consists in grouping terms in such a way that each group contains a common factor. To do this, sometimes it is necessary to use some artificial techniques.

Example 1.

x 4 – 3x 2 + 4x – 3 = 0.

Solution.

Let's imagine - 3x 2 = -2x 2 – x 2 and group:

(x 4 – 2x 2) – (x 2 – 4x + 3) = 0.

(x 4 – 2x 2 +1 – 1) – (x 2 – 4x + 3 + 1 – 1) = 0.

(x 2 – 1) 2 – 1 – (x – 2) 2 + 1 = 0.

(x 2 – 1) 2 – (x – 2) 2 = 0.

(x 2 – 1 – x + 2)(x 2 – 1 + x - 2) = 0.

(x 2 – x + 1)(x 2 + x – 3) = 0.

x 2 – x + 1 = 0 or x 2 + x – 3 = 0.

Answer: There are no roots in the first equation, from the second: x 1, 2 = (-1 ± √13)/2.

3. Factorization by the method of undetermined coefficients

The essence of the method is that the original polynomial is factorized with unknown coefficients. Using the property that polynomials are equal if their coefficients are equal at the same powers, the unknown expansion coefficients are found.

Example 1.

x 3 + 4x 2 + 5x + 2 = 0.

Solution.

A polynomial of degree 3 can be expanded into the product of linear and quadratic factors.

x 3 + 4x 2 + 5x + 2 = (x – a)(x 2 + bx + c),

x 3 + 4x 2 + 5x + 2 = x 3 +bx 2 + cx – ax 2 – abx – ac,

x 3 + 4x 2 + 5x + 2 = x 3 + (b – a)x 2 + (cx – ab)x – ac.

Having solved the system:

(b – a = 4,
(c – ab = 5,
(-ac = 2,

(a = -1,
(b = 3,
(c = 2, i.e.

x 3 + 4x 2 + 5x + 2 = (x + 1)(x 2 + 3x + 2).

The roots of the equation (x + 1)(x 2 + 3x + 2) = 0 are easy to find.

Answer: -1; -2.

4. Method of selecting a root using the highest and free coefficient

The method is based on the application of theorems:

1) Every integer root of a polynomial with integer coefficients is a divisor of the free term.

2) In order for the irreducible fraction p/q (p is an integer, q is a natural number) to be the root of an equation with integer coefficients, it is necessary that the number p be an integer divisor of the free term a 0, and q be a natural divisor of the leading coefficient.

Example 1.

6x 3 + 7x 2 – 9x + 2 = 0.

Solution:

6: q = 1, 2, 3, 6.

Therefore, p/q = ±1, ±2, ±1/2, ±1/3, ±2/3, ±1/6.

Having found one root, for example – 2, we will find other roots using corner division, the method of indefinite coefficients or Horner’s scheme.

Answer: -2; 1/2; 1/3.

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When solving algebraic equations, you often have to factor a polynomial. To factor a polynomial means to represent it as a product of two or more polynomials. We use some methods of decomposing polynomials quite often: taking a common factor, using abbreviated multiplication formulas, isolating a complete square, grouping. Let's look at some more methods.

Sometimes the following statements are useful when factoring a polynomial:

1) if a polynomial with integer coefficients has a rational root (where is an irreducible fraction, then is the divisor of the free term and the divisor of the leading coefficient:

2) If you somehow select the root of a polynomial of degree, then the polynomial can be represented in the form where is a polynomial of degree

A polynomial can be found either by dividing the polynomial into a binomial in a “column”, or by appropriately grouping the terms of the polynomial and separating the multiplier from them, or by the method of indefinite coefficients.

Example. Factor a polynomial

Solution. Since the coefficient of x4 is equal to 1, then the rational roots of this polynomial exist and are divisors of the number 6, i.e. they can be integers ±1, ±2, ±3, ±6. Let us denote this polynomial by P4(x). Since P P4 (1) = 4 and P4(-4) = 23, the numbers 1 and -1 are not roots of the polynomial PA(x). Since P4(2) = 0, then x = 2 is the root of the polynomial P4(x), and, therefore, this polynomial is divisible by the binomial x - 2. Therefore x4 -5x3 +7x2 -5x +6 x-2 x4 -2x3 x3 -3x2 +x-3

3x3 +7x2 -5x +6

3x3 +6x2 x2 - 5x + 6 x2- 2x

Therefore, P4(x) = (x - 2)(x3 - 3x2 + x - 3). Since xz - 3x2 + x - 3 = x2 (x - 3) + (x - 3) = (x - 3)(x2 + 1), then x4 - 5x3 + 7x2 - 5x + 6 = (x - 2) (x - 3)(x2 + 1).

Parameter input method

Sometimes when factoring a polynomial, the method of introducing a parameter helps. We will explain the essence of this method using the following example.

Example. x3 –(√3 + 1) x2 + 3.

Solution. Consider a polynomial with parameter a: x3 - (a + 1)x2 + a2, which at a = √3 turns into a given polynomial. Let's write this polynomial as a square trinomial for a: a - ax2 + (x3 - x2).

Since the roots of this trinomial squared with respect to a are a1 = x and a2 = x2 - x, then the equality a2 - ax2 + (xs - x2) = (a - x)(a - x2 + x) is true. Consequently, the polynomial x3 - (√3 + 1)x2 + 3 is decomposed into factors √3 – x and √3 - x2 + x, i.e.

x3 – (√3+1)x2+3=(x-√3)(x2-x-√3).

Method of introducing a new unknown

In some cases, by replacing the expression f(x) included in the polynomial Pn(x), through y one can obtain a polynomial with respect to y, which can easily be factorized. Then, after replacing y with f(x), we obtain a factorization of the polynomial Pn(x).

Example. Factor the polynomial x(x+1)(x+2)(x+3)-15.

Solution. Let's transform this polynomial as follows: x(x+1)(x+2)(x+3)-15= [x (x + 3)][(x + 1)(x + 2)] - 15 =(x2 + 3x)(x2 + 3x + 2) - 15.

Let's denote x2 + 3x by y. Then we have y(y + 2) - 15 = y2 + 2y - 15 = y2 + 2y + 1 - 16 = (y + 1)2 - 16 = (y + 1 + 4)(y + 1 - 4)= ( y+ 5)(y - 3).

Therefore x(x + 1)(x+ 2)(x + 3) - 15 = (x2+ 3x + 5)(x2 + 3x - 3).

Example. Factor the polynomial (x-4)4+(x+2)4

Solution. Let's denote x- 4+x+2 = x - 1 by y.

(x - 4)4 + (x + 2)2= (y - 3)4 + (y + 3)4 = y4 - 12y3 +54y3 - 108y + 81 + y4 + 12y3 + 54y2 + 108y + 81 =

2y4 + 108y2 + 162 = 2(y4 + 54y2 + 81) = 2[(yg + 27)2 - 648] = 2 (y2 + 27 - √b48)(y2 + 27+√b48)=

2((x-1)2+27-√b48)((x-1)2+27+√b48)=2(x2-2x + 28- 18√ 2)(x2- 2x + 28 + 18√ 2 ).

Combining different methods

Often, when factoring a polynomial, it is necessary to apply several of the methods discussed above in succession.

Example. Factor the polynomial x4 - 3x2 + 4x-3.

Solution. Using grouping, we rewrite the polynomial in the form x4 - 3x2 + 4x - 3 = (x4 – 2x2) – (x2 -4x + 3).

Applying the method of isolating a complete square to the first bracket, we have x4 - 3x3 + 4x - 3 = (x4 - 2 · 1 · x2 + 12) - (x2 -4x + 4).

Using the perfect square formula, we can now write that x4 – 3x2 + 4x - 3 = (x2 -1)2 - (x - 2)2.

Finally, applying the difference of squares formula, we get that x4 - 3x2 + 4x - 3 = (x2 - 1 + x - 2)(x2 - 1 - x + 2) = (x2+x-3)(x2 -x + 1 ).

§ 2. Symmetric equations

1. Symmetric equations of the third degree

Equations of the form ax3 + bx2 + bx + a = 0, a ≠ 0 (1) are called symmetric equations of the third degree. Since ax3 + bx2 + bx + a = a(x3 + 1) + bx (x + 1) = (x+1)(ax2+(b-a)x+a), then equation (1) is equivalent to the set of equations x + 1 = 0 and ax2 + (b-a)x + a = 0, which is not difficult to solve.

Example 1: Solve the equation

3x3 + 4x2 + 4x + 3 = 0. (2)

Solution. Equation (2) is a symmetric equation of the third degree.

Since 3x3 + 4xr + 4x + 3 = 3(x3 + 1) + 4x(x + 1) = (x+ 1)(3x2 - 3x + 3 + 4x) = (x + 1)(3x2 + x + 3) , then equation (2) is equivalent to the set of equations x + 1 = 0 and 3x3 + x +3=0.

The solution to the first of these equations is x = -1, the second equation has no solutions.

Answer: x = -1.

2. Symmetric equations of the fourth degree

Equation of the form

(3) is called a symmetric equation of the fourth degree.

Since x = 0 is not a root of equation (3), then by dividing both sides of equation (3) by x2, we obtain an equation equivalent to the original one (3):

Let us rewrite equation (4) as:

Let's make a substitution in this equation, then we get a quadratic equation

If equation (5) has 2 roots y1 and y2, then the original equation is equivalent to a set of equations

If equation (5) has one root y0, then the original equation is equivalent to the equation

Finally, if equation (5) has no roots, then the original equation also has no roots.

Example 2: Solve the equation

Solution. This equation is a symmetric equation of the fourth degree. Since x = 0 is not its root, then by dividing equation (6) by x2, we obtain an equivalent equation:

Having grouped the terms, we rewrite equation (7) in the form or in the form

Putting it, we get an equation that has two roots y1 = 2 and y2 = 3. Consequently, the original equation is equivalent to a set of equations

The solution to the first equation of this set is x1 = 1, and the solution to the second is u.

Therefore, the original equation has three roots: x1, x2 and x3.

Answer: x1=1.

§3. Algebraic equations

1. Reducing the degree of the equation

Some algebraic equations, by replacing a certain polynomial in them with one letter, can be reduced to algebraic equations whose degree is less than the degree of the original equation and whose solution is simpler.

Example 1: Solve the equation

Solution. Let us denote by, then equation (1) can be rewritten as The last equation has roots and Therefore, equation (1) is equivalent to the set of equations and. The solution to the first equation of this set is and the solution to the second equation is

The solutions to equation (1) are

Example 2: Solve the equation

Solution. Multiplying both sides of the equation by 12 and denoting by,

We obtain the equation. We rewrite this equation in the form

(3) and denoting by we rewrite equation (3) in the form The last equation has roots and Therefore, we obtain that equation (3) is equivalent to a set of two equations and There are solutions to this set of equations and i.e. equation (2) is equivalent to a set of equations and ( 4)

The solutions to the set (4) are and, and they are the solutions to equation (2).

2. Equations of the form

The equation

(5) where are the given numbers, can be reduced to a biquadratic equation by replacing the unknown, i.e., replacing

Example 3: Solve the equation

Solution. Let us denote by,t. e. we make a change of variables or Then equation (6) can be rewritten in the form or, using the formula, in the form

Since the roots of a quadratic equation are and, the solutions to equation (7) are solutions to the set of equations and. This set of equations has two solutions and Therefore, the solutions to equation (6) are and

3. Equations of the form

The equation

(8) where the numbers α, β, γ, δ, and Α are such that α

Example 4: Solve the equation

Solution. Let's make a change of unknowns, i.e. y=x+3 or x = y – 3. Then equation (9) can be rewritten as

(y-2)(y-1)(y+1)(y+2)=10, i.e. in the form

(y2- 4)(y2-1)=10(10)

Biquadratic equation (10) has two roots. Therefore, equation (9) also has two roots:

4. Equations of the form

Equation, (11)

Where, x = 0 has no root, therefore, dividing equation (11) by x2, we obtain an equivalent equation

Which, after replacing the unknown, will be rewritten in the form of a quadratic equation, the solution of which is not difficult.

Example 5: Solve the equation

Solution. Since h = 0 is not a root of equation (12), dividing it by x2, we obtain an equivalent equation

Making the replacement unknown, we get the equation (y+1)(y+2)=2, which has two roots: y1 = 0 and y1 = -3. Consequently, the original equation (12) is equivalent to the set of equations

This set has two roots: x1= -1 and x2 = -2.

Answer: x1= -1, x2 = -2.

Comment. Equation of the form

Which can always be reduced to the form (11) and, moreover, considering α > 0 and λ > 0 to the form.

5. Equations of the form

The equation

,(13) where the numbers, α, β, γ, δ, and Α are such that αβ = γδ ≠ 0, can be rewritten by multiplying the first bracket with the second, and the third with the fourth, in the form i.e. equation (13) is now written in the form (11), and its solution can be carried out in the same way as solving equation (11).

Example 6: Solve the equation

Solution. Equation (14) has the form (13), so we rewrite it in the form

Since x = 0 is not a solution to this equation, then by dividing both sides by x2, we obtain an equivalent original equation. Making a change of variables, we obtain a quadratic equation whose solution is and. Consequently, the original equation (14) is equivalent to the set of equations and.

The solution to the first equation of this set is

The second equation of this set of solutions has no solutions. So, the original equation has roots x1 and x2.

6. Equations of the form

The equation

(15) where the numbers a, b, c, q, A are such that x = 0 does not have a root, therefore, dividing equation (15) by x2. we obtain an equivalent equation, which, after replacing the unknown, will be rewritten in the form of a quadratic equation, the solution of which is not difficult.

Example 7. Solving the equation

Solution. Since x = 0 is not a root of equation (16), then dividing both sides by x2, we obtain the equation

, (17) equivalent to equation (16). Having made the replacement with an unknown, we rewrite equation (17) in the form

Quadratic equation (18) has 2 roots: y1 = 1 and y2 = -1. Therefore, equation (17) is equivalent to the set of equations and (19)

The set of equations (19) has 4 roots: ,.

They will be the roots of equation (16).

§4. Rational equations

Equations of the form = 0, where H(x) and Q(x) are polynomials, are called rational.

Having found the roots of the equation H(x) = 0, then you need to check which of them are not roots of the equation Q(x) = 0. These roots and only they will be solutions to the equation.

Let's consider some methods for solving equations of the form = 0.

1. Equations of the form

The equation

(1) under certain conditions on the numbers can be solved as follows. By grouping the terms of equation (1) by two and summing each pair, it is necessary to obtain in the numerator polynomials of the first or zero degree, differing only in numerical factors, and in the denominators - trinomials with the same two terms containing x, then after replacing the variables, the resulting equation will either have also form (1), but with a smaller number of terms, or will be equivalent to a set of two equations, one of which will be of the first degree, and the second will be an equation of type (1), but with a smaller number of terms.

Example. Solve the equation

Solution. Having grouped on the left side of equation (2) the first term with the last, and the second with the penultimate, we rewrite equation (2) in the form

Summing the terms in each bracket, we rewrite equation (3) in the form

Since there is no solution to equation (4), then, dividing this equation by, we obtain the equation

, (5) equivalent to equation (4). Let us make a replacement for the unknown, then equation (5) will be rewritten in the form

Thus, the solution to equation (2) with five terms on the left side is reduced to the solution to equation (6) of the same form, but with three terms on the left side. Summing up all the terms on the left side of equation (6), we rewrite it in the form

There are solutions to the equation. None of these numbers makes the denominator of the rational function on the left side of equation (7) vanish. Consequently, equation (7) has these two roots, and therefore the original equation (2) is equivalent to the set of equations

The solutions to the first equation of this set are

The solutions to the second equation from this set are

Therefore, the original equation has roots

2. Equations of the form

The equation

(8) under certain conditions on numbers can be solved as follows: it is necessary to select the integer part in each of the fractions of the equation, i.e. replace equation (8) with the equation

Reduce it to form (1) and then solve it in the manner described in the previous paragraph.

Example. Solve the equation

Solution. Let us write equation (9) in the form or in the form

Summing up the terms in brackets, we rewrite equation (10) in the form

By replacing the unknown, we rewrite equation (11) in the form

Summing up the terms on the left side of equation (12), we rewrite it in the form

It is easy to see that equation (13) has two roots: and. Therefore, the original equation (9) has four roots:

3) Equations of the form.

An equation of the form (14), under certain conditions for numbers, can be solved as follows: by expanding (if this is, of course, possible) each of the fractions on the left side of equation (14) into a sum of simple fractions

Reduce equation (14) to form (1), then, having carried out a convenient rearrangement of the terms of the resulting equation, solve it using the method described in paragraph 1).

Example. Solve the equation

Solution. Since and, then by multiplying the numerator of each fraction in equation (15) by 2 and noting that equation (15) can be written as

Equation (16) has the form (7). Having rearranged the terms in this equation, we rewrite it in the form or in the form

Equation (17) is equivalent to the set of equations and

To solve the second equation of the set (18), we make a replacement for the unknown. Then it will be rewritten in the form or in the form

Summing up all the terms on the left side of equation (19), rewrite it in the form

Since the equation has no roots, equation (20) also does not have them.

The first equation of the set (18) has a single root. Since this root is included in the ODZ of the second equation of the set (18), it is the only root of the set (18), and therefore of the original equation.

4. Equations of the form

The equation

(21) under certain conditions on the numbers and A after representing each term on the left side in the form can be reduced to the form (1).

Example. Solve the equation

Solution. Let us rewrite equation (22) in the form or in the form

Thus, equation (23) is reduced to form (1). Now, grouping the first term with the last, and the second with the third, we rewrite equation (23) in the form

This equation is equivalent to the set of equations and. (24)

The last equation of the set (24) can be rewritten as

There are solutions to this equation and, since it is included in the ODZ of the second equation of the set (30), the set (24) has three roots:. All of them are solutions to the original equation.

5. Equations of the form.

Equation of the form (25)

Under certain conditions on numbers, by replacing the unknown, one can reduce to an equation of the form

Example. Solve the equation

Solution. Since it is not a solution to equation (26), then dividing the numerator and denominator of each fraction on the left side by, we rewrite it in the form

Having made a change of variables, we rewrite equation (27) in the form

Solving equation (28) there is and. Therefore, equation (27) is equivalent to the set of equations and. (29)

Methods for solving equations: n n n Replacing the equation h(f(x)) = h(g(x)) with the equation f(x) = g(x) Factorization. Introduction of a new variable. Functional - graphic method. Selection of roots. Application of Vieta's formulas.

Replacing the equation h(f(x)) = h(g(x)) with the equation f(x) = g(x). The method can be used only in the case when y = h(x) is a monotonic function that takes each value once. If the function is non-monotonic, then loss of roots is possible.

Solve the equation (3 x + 2)²³ = (5 x – 9)²³ y = x ²³ is an increasing function, so from the equation (3 x + 2)²³ = (5 x – 9)²³ you can go to the equation 3 x + 2 = 5 x – 9, from where we find x = 5, 5. Answer: 5, 5.

Factorization. The equation f(x)g(x)h(x) = 0 can be replaced by a set of equations f(x) = 0; g(x) = 0; h(x) = 0. Having solved the equations of this set, you need to take those roots that belong to the domain of definition of the original equation, and discard the rest as extraneous.

Solve the equation x³ – 7 x + 6 = 0 Representing the term 7 x in the form x + 6 x, we obtain sequentially: x³ – x – 6 x + 6 = 0 x(x² – 1) – 6(x – 1) = 0 x (x – 1)(x + 1) – 6(x – 1) = 0 (x – 1)(x² + x – 6) = 0 Now the problem is reduced to solving a set of equations x – 1 = 0; x² + x – 6 = 0. Answer: 1, 2, – 3.

Introduction of a new variable. If the equation y(x) = 0 can be transformed to the form p(g(x)) = 0, then you need to introduce a new variable u = g(x), solve the equation p(u) = 0, and then solve the set of equations g( x) = u 1; g(x) = u 2; ... ; g(x) = un, where u 1, u 2, …, un are the roots of the equation p(u) = 0.

Solve the equation A special feature of this equation is the equality of the coefficients of its left side, equidistant from its ends. Such equations are called reciprocal. Since 0 is not a root of this equation, dividing by x² we get

Let's introduce a new variable. Then we get a quadratic equation. So the root y 1 = – 1 can be ignored. We get the Answer: 2, 0, 5.

Solve the equation 6(x² – 4)² + 5(x² – 4)(x² – 7 x +12) + (x² – 7 x + 12)² = 0 This equation can be solved as a homogeneous equation. Let's divide both sides of the equation by (x² – 7 x +12)² (it is clear that the values ​​of x are such that x² – 7 x +12=0 are not solutions). Now we denote We Have From Here Answer:

Functional - graphic method. If one of the functions y = f(x), y = g(x) increases, and the other decreases, then the equation f(x) = g(x) either has no roots or has one root.

Solve the equation It is fairly obvious that x = 2 is the root of the equation. Let us prove that this is the only root. Let's transform the equation to the form We notice that the function increases, and the function decreases. This means that the equation has only one root. Answer: 2.

Selection of roots n n n Theorem 1: If an integer m is the root of a polynomial with integer coefficients, then the free term of the polynomial is divisible by m. Theorem 2: The reduced polynomial with integer coefficients has no fractional roots. Theorem 3: – equation with integer Let coefficients. If a number and a fraction where p and q are irreducible integers is the root of an equation, then p is a divisor of the free term an, and q is a divisor of the coefficient of the leading term a 0.

Bezout's theorem. The remainder when dividing any polynomial by a binomial (x – a) is equal to the value of the polynomial being divided at x = a. Corollaries of Bezout's theorem n n n n The difference of identical powers of two numbers is divided without a remainder by the difference of the same numbers; The difference between identical even powers of two numbers is divided without a remainder by both the difference of these numbers and their sum; The difference between identical odd powers of two numbers is not divisible by the sum of these numbers; The sum of equal powers of two non-numbers is divided by the difference of these numbers; The sum of identical odd powers of two numbers is divided without a remainder by the sum of these numbers; The sum of identical even powers of two numbers is not divisible either by the difference of these numbers or by their sum; A polynomial is divisible by a binomial (x – a) if and only if the number a is the root of the given polynomial; The number of distinct roots of a nonzero polynomial is no more than its degree.

Solve the equation x³ – 5 x² – x + 21 = 0 The polynomial x³ – 5 x² – x + 21 has integer coefficients. By Theorem 1, its integer roots, if any, are among the divisors of the free term: ± 1, ± 3, ± 7, ± 21. By checking we are convinced that the number 3 is a root. By corollary to Bezout's theorem, the polynomial is divisible by (x – 3). Thus, x³– 5 x² – x + 21 = (x – 3)(x²– 2 x – 7). Answer:

Solve the equation 2 x³ – 5 x² – x + 1 = 0 According to Theorem 1, only numbers ± 1 can be integer roots of the equation. Checking shows that these numbers are not roots. Since the equation is not reduced, it can have fractional rational roots. Let's find them. To do this, multiply both sides of the equation by 4: 8 x³ – 20 x² – 4 x + 4 = 0 Substituting 2 x = t, we get t³ – 5 t² – 2 t + 4 = 0. By Theorem 2, all rational roots of this given equation must be intact. They can be found among the divisors of the free term: ± 1, ± 2, ± 4. In this case, t = – 1 is suitable. Therefore, by corollary to Bezout’s theorem, the polynomial 2 x³ – 5 x² – x + 1 is divisible by (x + 0, 5 ): 2 x³ – 5 x² – x + 1 = (x + 0. 5)(2 x² – 6 x + 2) Having solved the quadratic equation 2 x² – 6 x + 2 = 0, we find the remaining roots: Answer:

Solve the equation 6 x³ + x² – 11 x – 6 = 0 According to Theorem 3, the rational roots of this equation should be sought among the numbers. Substituting them one by one into the equation, we find that they satisfy the equation. They exhaust all the roots of the equation. Answer:

Find the sum of the squared roots of the equation x³ + 3 x² – 7 x +1 = 0 By Vieta’s theorem Note that where

Indicate how each of these equations can be solved. Solve equations No. 1, 4, 15, 17.

Answers and directions: 1. Introduction of a new variable. 2. Functional - graphic method. 3. Replacing the equation h(f(x)) = h(g(x)) with the equation f(x) = g(x). 4. Factorization. 5. Selection of roots. 6 Functional - graphic method. 7. Application of Vieta formulas. 8. Selection of roots. 9. Replacing the equation h(f(x)) = h(g(x)) with the equation f(x) = g(x). 10. Introduction of a new variable. 11. Factorization. 12. Introduction of a new variable. 13. Selection of roots. 14. Application of Vieta formulas. 15. Functional - graphic method. 16. Factorization. 17. Introduction of a new variable. 18. Factorization.

1. Instruction. Write the equation as 4(x²+17 x+60)(x+16 x+60)=3 x², Divide both sides by x². Enter the variable Answer: x 1 = – 8; x 2 = – 7.5. 4. Instruction. Add 6 y and – 6 y to the left side of the equation and write it as (y³ – 2 y²) + (– 3 y² + 6 y) + (– 8 y + 16) = (y – 2)(y² – 3 y - 8). Answer:

14. Instruction. According to Vieta's theorem Since these are integers, the roots of the equation can only be the numbers – 1, – 2, – 3. Answer: 15. Answer: – 1. 17. Instruction. Divide both sides of the equation by x² and write it as Enter a variable Answer: 1; 15; 2; 3.

Bibliography. n n n Kolmogorov A. N. “Algebra and the beginnings of analysis, 10 – 11” (M.: Prosveshchenie, 2003). Bashmakov M. I. “Algebra and the beginnings of analysis, 10 – 11” (M.: Prosveshchenie, 1993). Mordkovich A. G. “Algebra and principles of analysis, 10 – 11” (M.: Mnemosyna, 2003). Alimov Sh. A., Kolyagin Yu. M. et al. “Algebra and the beginnings of analysis, 10 – 11” (M.: Prosveshchenie, 2000). Galitsky M. L., Goldman A. M., Zvavich L. I. “Collection of problems in algebra, 8 – 9” (M.: Prosveshchenie, 1997). Karp A. P. “Collection of problems on algebra and principles of analysis, 10 – 11” (M.: Prosveshchenie, 1999). Sharygin I. F. “Optional course in mathematics, problem solving, 10” (M.: Education. 1989). Skopets Z. A. “Additional chapters on the course of mathematics, 10” (M.: Prosveshchenie, 1974). Litinsky G.I. “Mathematics Lessons” (Moscow: Aslan, 1994). Muravin G.K. “Equations, inequalities and their systems” (Mathematics, supplement to the newspaper “First of September”, No. 2, 3, 2003). Kolyagin Yu. M. “Polynomials and equations of higher degrees” (Mathematics, supplement to the newspaper “First of September”, No. 3, 2005).

Basic goals:

1. Reinforce the concept of an entire rational equation of the th degree.
2. Formulate the basic methods for solving equations of higher degrees (n > 3).
3. Teach basic methods for solving higher-order equations.
4. To teach how to determine the most effective method his decisions.

Forms, methods and pedagogical techniques used by the teacher in the classroom:

• Lecture-seminar teaching system (lectures - explanation of new material, seminars - problem solving).
• Information and communication technologies (frontal survey, oral work with the class).
• Differentiated learning, group and individual forms.
• Using a research method in teaching aimed at developing the mathematical apparatus and thinking abilities of each individual student.
• Printed material – individual short summary lesson (basic concepts, formulas, statements, lecture material condensed in the form of diagrams or tables).

Lesson plan:

1. Organizing time.
   The purpose of the stage: to include students in educational activities, determine the content framework of the lesson.
2. Updating students' knowledge.
   The purpose of the stage: to update students’ knowledge on previously studied related topics
3. Studying new topic(lecture). Goal of the stage: to formulate the basic methods for solving equations of higher degrees (n > 3)
4. Summarizing.
   The purpose of the stage: to once again highlight the key points in the material studied in the lesson.
5. Homework.
   The purpose of the stage: to formulate homework for students.

Lesson summary

1. Organizational moment.

Formulation of the lesson topic: “Equations of higher powers. Methods for solving them.”

2. Updating students' knowledge.

Theoretical survey - conversation. Repetition of some previously studied information from the theory. Students formulate basic definitions and formulate the necessary theorems. Give examples to demonstrate the level of previously acquired knowledge.

• The concept of an equation with one variable.
• The concept of the root of an equation, the solution of an equation.
• Concept linear equation with one variable, the concept of a quadratic equation with one variable.
• The concept of equivalence of equations, equations-consequences (the concept of extraneous roots), transition not by consequence (the case of loss of roots).
• The concept of a whole rational expression with one variable.
• The concept of a whole rational equation n th degree. Standard form of a whole rational equation. Reduced whole rational equation.
• Transition to a set of equations of lower degrees by factoring the original equation.
• The concept of a polynomial n th degree from x. Bezout's theorem. Corollaries from Bezout's theorem. Root theorems ( Z-roots and Q-roots) of an entire rational equation with integer coefficients (reduced and unreduced, respectively).
• Horner's scheme.

3. Studying a new topic.

We will consider the whole rational equation n-th power of standard form with one unknown variable x:Pn(x)= 0, where P n (x) = a n x n + a n-1 x n-1 + a 1 x + a 0– polynomial n th degree from x, a n ≠ 0. If a n = 1 then such an equation is called a reduced integer rational equation n th degree. Let us consider such equations for different meanings n and list the main methods for solving them.

n= 1 – linear equation.

n= 2 – quadratic equation. Discriminant formula. Formula for calculating roots. Vieta's theorem. Selecting a complete square.

n= 3 – cubic equation.

Grouping method.

Example: x 3 – 4x 2 – x+ 4 = 0 (x – 4)(x 2– 1) = 0 x 1 = 4 , x 2 = 1,x 3 = -1.

Reciprocal cubic equation of the form ax 3 + bx 2 + bx + a= 0. We solve by combining terms with the same coefficients.

Example: x 3 – 5x 2 – 5x + 1 = 0 (x + 1)(x 2 – 6x + 1) = 0 x 1 = -1, x 2 = 3 + 2, x 3 = 3 – 2.

Selection of Z-roots based on the theorem. Horner's scheme. When applying this method, it is necessary to emphasize that the search in this case is finite, and we select the roots using a certain algorithm in accordance with the theorem Z-roots of the given whole rational equation with integer coefficients.

Example: x 3 – 9x 2 + 23x– 15 = 0. The equation is given. Let us write down the divisors of the free term ( + 1; + 3; + 5; + 15). Let's apply Horner's scheme:

	x 3	x 2	x 1	x 0	conclusion
	1	-9	23	-15
1	1	1 x 1 – 9 = -8	1 x (-8) + 23 = 15	1 x 15 – 15 = 0	1 – root
	x 2	x 1	x 0

We get ( x – 1)(x 2 – 8x + 15) = 0 x 1 = 1, x 2 = 3, x 3 = 5.

Equation with integer coefficients. Selection of Q-roots based on the theorem. Horner's scheme. When applying this method, it is necessary to emphasize that the search in this case is finite and we select the roots using a certain algorithm in accordance with the theorem about Q-roots of an unreduced integer rational equation with integer coefficients.

Example: 9 x 3 + 27x 2 – x– 3 = 0. The equation is unreduced. Let us write down the divisors of the free term ( + 1; + 3). Let us write down the divisors of the coefficient at the highest power of the unknown. ( + 1; + 3; + 9) Consequently, we will look for roots among the values ​​( + 1; + ; + ; + 3). Let's apply Horner's scheme:

	x 3	x 2	x 1	x 0	conclusion
	9	27	-1	-3
1	9	1 x 9 + 27 = 36	1 x 36 – 1 = 35	1 x 35 – 3 = 32 ≠ 0	1 – not a root
-1	9	-1 x 9 + 27 = 18	-1 x 18 – 1 = -19	-1 x (-19) – 3 = 16 ≠ 0	-1 – not a root
	9	x 9 + 27 = 30	x 30 – 1 = 9	x 9 – 3 = 0	root
	x 2	x 1	x 0

We get ( x – )(9x 2 + 30x + 9) = 0 x 1 = , x 2 = - , x 3 = -3.

For ease of calculation when selecting Q -roots It can be convenient to make a change of variable, go to the given equation and select Z -roots.

• If the dummy term is 1

.

• If you can use a replacement of the form y = kx

.

Cardano formula. There is a universal method for solving cubic equations - this is the Cardano formula. This formula is associated with the names of Italian mathematicians Gerolamo Cardano (1501–1576), Nicolo Tartaglia (1500–1557), and Scipione del Ferro (1465–1526). This formula is beyond the scope of our course.

n= 4 – equation of the fourth degree.

Grouping method.

Example: x 4 + 2x 3 + 5x 2 + 4x – 12 = 0 (x 4 + 2x 3) + (5x 2 + 10x) – (6x + 12) = 0 (x + 2)(x 3 + 5x – 6) = 0 (x + 2)(x– 1)(x 2 + x + 6) = 0 x 1 = -2, x 2 = 1.

Variable replacement method.

• Biquadratic equation of the form ax 4 + bx 2 + s = 0 .

Example: x 4 + 5x 2 – 36 = 0. Replacement y = x 2. From here y 1 = 4, y 2 = -9. That's why x 1,2 = + 2 .

• Reciprocal equation of the fourth degree of the form ax 4 + bx 3+c x 2 + bx + a = 0.

We solve by combining terms with the same coefficients by replacing the form

• ax 4 + bx 3 + cx 2 – bx + a = 0.

• Generalized recurrent equation of the fourth degree of the form ax 4 + bx 3 + cx 2 + kbx + k 2 a = 0.

• Replacement general view. Some standard replacements.

Example 3 . General view replacement(follows from the type of specific equation).

n = 3.

Equation with integer coefficients. Selection of Q-roots n = 3.

General formula. There is a universal method for solving fourth degree equations. This formula is associated with the name of Ludovico Ferrari (1522–1565). This formula is beyond the scope of our course.

n > 5 – equations of the fifth and higher degrees.

Equation with integer coefficients. Selection of Z-roots based on the theorem. Horner's scheme. The algorithm is similar to that discussed above for n = 3.

Equation with integer coefficients. Selection of Q-roots based on the theorem. Horner's scheme. The algorithm is similar to that discussed above for n = 3.

Symmetric equations. Any reciprocal equation of odd degree has a root x= -1 and after factoring it into factors we find that one factor has the form ( x+ 1), and the second factor is a reciprocal equation of even degree (its degree is one less than the degree of the original equation). Any reciprocal equation of even degree together with a root of the form x = φ also contains the root of the species. Using these statements, we solve the problem by lowering the degree of the equation under study.

Variable replacement method. Use of homogeneity.

There is no general formula for solving entire equations of the fifth degree (this was shown by the Italian mathematician Paolo Ruffini (1765–1822) and the Norwegian mathematician Niels Henrik Abel (1802–1829)) and higher degrees (this was shown by the French mathematician Evariste Galois (1811–1832) )).

• Let us recall once again that in practice it is possible to use combinations the methods listed above. It is convenient to pass to a set of equations of lower degrees by factoring the original equation.
• Outside the scope of our discussion today are those widely used in practice. graphical methods solving equations and approximate solution methods equations of higher degrees.
• There are situations when the equation does not have R-roots.
Then the solution comes down to showing that the equation has no roots. To prove this, we analyze the behavior of the functions under consideration on monotonicity intervals. Example: equation x 8 – x 3 + 1 = 0 has no roots.
• Using the property of monotonicity of functions
. There are situations when using various properties of functions allows you to simplify the task.
Example 1: Equation x 5 + 3x– 4 = 0 has one root x= 1. Due to the property of monotonicity of the analyzed functions, there are no other roots.
Example 2: Equation x 4 + (x– 1) 4 = 97 has roots x 1 = -2 and x 2 = 3. Having analyzed the behavior of the corresponding functions on intervals of monotonicity, we conclude that there are no other roots.

4. Summing up.

Summary: Now we have mastered the basic methods for solving various equations of higher degrees (for n > 3). Our task is to learn how to effectively use the algorithms listed above. Depending on the type of equation, we will have to learn to determine which method of solution in a given case is the most effective, as well as correctly apply the chosen method.

5. Homework.

: paragraph 7, pp. 164–174, nos. 33–36, 39–44, 46.47.

: №№ 9.1–9.4, 9.6–9.8, 9.12, 9.14–9.16, 9.24–9.27.

Possible topics for reports or abstracts on this topic:

• Cardano formula
• Graphical method for solving equations. Examples of solutions.
• Methods for approximate solution of equations.

Analysis of student learning and interest in the topic:

Experience shows that students’ interest is primarily aroused by the possibility of selecting Z-roots and Q-roots of equations using a fairly simple algorithm using Horner’s scheme. Students are also interested in various standard types of substitution of variables, which can significantly simplify the type of problem. Graphical solution methods are usually of particular interest. In this case, you can additionally analyze problems using a graphical method for solving equations; discuss the general form of the graph for a polynomial of degree 3, 4, 5; analyze how the number of roots of equations of degree 3, 4, 5 is related to the appearance of the corresponding graph. Below is a list of books where you can find additional information on this topic.

Bibliography:

1. Vilenkin N.Ya. and others. “Algebra. Textbook for 9th grade students in-depth study Mathematics” – M., Education, 2007 – 367 p.
2. Vilenkin N.Ya., Shibasov L.P., Shibasova Z.F.“Behind the pages of a mathematics textbook. Arithmetic. Algebra. 10-11 grades” – M., Education, 2008 – 192 p.
3. Vygodsky M.Ya.“Handbook of Mathematics” – M., AST, 2010 – 1055 p.
4. Galitsky M.L.“Collection of problems in algebra. Textbook for grades 8-9 with in-depth study of mathematics” - M., Prosveshchenie, 2008 - 301 p.
5. Zvavich L.I. and others. “Algebra and the beginnings of analysis. 8–11 grades A manual for schools and classes with in-depth study of mathematics” - M., Drofa, 1999 - 352 p.
6. Zvavich L.I., Averyanov D.I., Pigarev B.P., Trushanina T.N.“Mathematics assignments for preparing for the written exam in 9th grade” - M., Prosveshchenie, 2007 - 112 p.
7. Ivanov A.A., Ivanov A.P.“Thematic tests for systematizing knowledge in mathematics” part 1 – M., Fizmatkniga, 2006 – 176 p.
8. Ivanov A.A., Ivanov A.P.“Thematic tests for systematizing knowledge in mathematics” part 2 – M., Fizmatkniga, 2006 – 176 p.
9. Ivanov A.P.“Tests and test papers mathematics. Tutorial". – M., Fizmatkniga, 2008 – 304 p.
10. Leibson K.L.“Collection of practical tasks in mathematics. Part 2–9 grades” – M., MTSNM, 2009 – 184 p.
11. Makarychev Yu.N., Mindyuk N.G."Algebra. Additional chapters to school textbook 9th grade. A textbook for students in schools and classes with in-depth study of mathematics.” – M., Education, 2006 – 224 p.
12. Mordkovich A.G."Algebra. In-depth study. 8th grade. Textbook” – M., Mnemosyne, 2006 – 296 p.
13. Savin A.P. “encyclopedic Dictionary young mathematician” – M., Pedagogy, 1985 – 352 p.
14. Survillo G.S., Simonov A.S.“Didactic materials on algebra for grade 9 with in-depth study of mathematics” - M., Prosveshchenie, 2006 - 95 p.
15. Chulkov P.V.“Equations and inequalities in school course mathematician. Lectures 1–4” – M., September 1st, 2006 – 88 p.
16. Chulkov P.V.“Equations and inequalities in the school mathematics course. Lectures 5–8” – M., September 1, 2009 – 84 p.

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Introduction

Solving algebraic equations of higher degrees with one unknown is one of the most difficult and ancient mathematical problems. The most outstanding mathematicians of antiquity dealt with these problems.

Solving equations of the nth degree is an important task for modern mathematics. There is quite a lot of interest in them, since these equations are closely related to the search for the roots of equations that are not covered in the school mathematics curriculum.

Problem: Students’ lack of skills in solving equations of higher degrees in various ways prevents them from successfully preparing for final certification in mathematics and mathematical Olympiads, and training in a specialized mathematics class.

The listed facts determined relevance our work “Solving equations of higher degrees”.

Knowledge of the simplest methods of solving equations of the nth degree reduces the time for completing a task, on which the result of the work and the quality of the learning process depend.

Goal of the work: studying known methods solving equations of higher degrees and identifying the most accessible of them for practical application.

Based on the goal, the following is defined in the work: tasks:

Study literature and Internet resources on this topic;

Get acquainted with historical facts related to this topic;

Describe different ways to solve higher degree equations

compare the degree of complexity of each of them;

Introduce classmates to ways of solving equations of higher degrees;

Create a selection of equations for the practical application of each of the considered methods.

Object of study- equations of higher degrees with one variable.

Subject of study- methods for solving equations of higher degrees.

Hypothesis: There is no general method or single algorithm that allows one to find solutions to equations of the nth degree in a finite number of steps.

Research methods:

- bibliographic method (analysis of literature on the research topic);

- classification method;

- method of qualitative analysis.

Theoretical significance research consists of systematizing methods for solving equations of higher degrees and describing their algorithms.

Practical significance- presented material on this topic and development teaching aid for students on this topic.

1. EQUATIONS OF HIGHER DEGREES

1.1 Concept of nth degree equation

Definition 1. An equation of the nth degree is an equation of the form

a 0 xⁿ+a 1 x n -1 +a 2 xⁿ - ²+…+a n -1 x+a n = 0, where coefficients a 0, a 1, a 2…, a n -1, a n- any real numbers, and ,a 0 ≠ 0 .

Polynomial a 0 xⁿ+a 1 x n -1 +a 2 xⁿ - ²+…+a n -1 x+a n is called a polynomial of nth degree. The coefficients are distinguished by names: a 0 - senior coefficient; a n is a free member.

Definition 2. Solutions or roots for a given equation are all the values ​​of the variable X, which turn this equation into a true numerical equality or, for which the polynomial a 0 xⁿ+a 1 x n -1 +a 2 xⁿ - ²+…+a n -1 x+a n goes to zero. This variable value X also called the root of a polynomial. Solving an equation means finding all its roots or establishing that there are none.

If a 0 = 1, then such an equation is called a reduced integer rational equation n th degrees.

For equations of the third and fourth degree, there are Cardano and Ferrari formulas that express the roots of these equations through radicals. It turned out that in practice they are rarely used. Thus, if n ≥ 3, and the coefficients of the polynomial are arbitrary real numbers, then finding the roots of the equation is not an easy task. However, in many special cases this problem is completely solved. Let's look at some of them.

1.2 Historical facts solving higher degree equations

Already in ancient times, people realized how important it was to learn to solve algebraic equations. About 4000 years ago, Babylonian scientists knew how to solve a quadratic equation and solved systems of two equations, one of which was of the second degree. With the help of equations of higher degrees, various problems of land surveying, architecture and military affairs were solved; many and varied questions of practice and natural science were reduced to them, since the precise language of mathematics allows one to simply express facts and relationships, which, when stated in ordinary language, may seem confusing and complex .

A universal formula for finding the roots of an algebraic equation nth no degree. Many, of course, had the tempting idea of ​​finding, for any degree n, formulas that would express the roots of the equation through its coefficients, that is, solve the equation in radicals.

Only in the 16th century did Italian mathematicians manage to advance further - to find formulas for n = 3 and n = 4. At the same time, Scipio, Dahl, Ferro and his students Fiori and Tartaglia were studying the question of the general solution of equations of the 3rd degree.

In 1545, the book of the Italian mathematician D. Cardano “Great Art, or on the Rules of Algebra” was published, where, along with other questions of algebra, general methods for solving cubic equations are considered, as well as a method for solving equations of the 4th degree, discovered by his student L. Ferrari.

A complete presentation of issues related to the solution of equations of the 3rd and 4th degrees was given by F. Viet.

In the 20s of the 19th century, the Norwegian mathematician N. Abel proved that the roots of equations of the fifth degree cannot be expressed in terms of radicals.

The study revealed that modern science There are many ways to solve equations of the nth degree.

The result of the search for methods for solving equations of higher degrees that cannot be solved by the methods considered in school curriculum, methods based on the application of Vieta’s theorem (for equations of degree n>2), Bezout's theorems, Horner's schemes, as well as the Cardano and Ferrari formula for solving cubic and quartic equations.

The work presents methods for solving equations and their types, which became a discovery for us. These include the method of indefinite coefficients, the selection of the full degree, symmetric equations.

2. SOLUTION OF ENTIRE EQUATIONS OF HIGHER DEGREES WITH INTEGER COEFFICIENTS

2.1 Solving 3rd degree equations. Formula D. Cardano

Consider equations of the form x 3 +px+q=0. Let us transform the general equation to the form: x 3 +px 2 +qx+r=0. Let's write down the formula for the cube of the sum; Let's add it to the original equality and replace it with y. We get the equation: y 3 + (q -) (y -) + (r - =0. After transformations, we have: y 2 +py + q=0. Now, let’s write down the sum cube formula again:

(a+b) 3 =a 3 + 3a 2 b + 3ab 2 +b 3 = a 3 +b 3 + 3ab (a + b), replace ( a+b)on x, we get the equation x 3 - 3abx - (a 3 +b 3) = 0. Now we can see that the original equation is equivalent to the system: and Solving the system, we get:

We have obtained a formula for solving the above 3rd degree equation. It bears the name of the Italian mathematician Cardano.

Let's look at an example. Solve the equation: .

We have R= 15 and q= 124, then using the Cardano formula we calculate the root of the equation

Conclusion: this formula good, but not suitable for solving all cubic equations. At the same time, it is cumbersome. Therefore, in practice it is rarely used.

But anyone who masters this formula can use it when solving third-degree equations on the Unified State Exam.

2.2 Vieta's theorem

From a mathematics course we know this theorem for a quadratic equation, but few people know that it is also used to solve higher-order equations.

Consider the equation:

let's break it down left side equations into factors, divide by ≠ 0.

Let's transform the right side of the equation to the form

; It follows from this that we can write the following equalities into the system:

The formulas derived by Viète for quadratic equations and demonstrated by us for equations of the 3rd degree are also true for polynomials of higher degrees.

Let's solve the cubic equation:

Conclusion: this method universal and easy enough for students to understand, since Vieta’s theorem is familiar to them from the school curriculum for n = 2. At the same time, in order to find the roots of equations using this theorem, you must have good computational skills.

2.3 Bezout's theorem

This theorem is named after the 18th century French mathematician J. Bezout.

Theorem. If the equation a 0 xⁿ+a 1 x n -1 +a 2 xⁿ - ²+…+a n -1 x+a n = 0, in which all coefficients are integers, and the free term is non-zero and has an integer root, then this root is a divisor of the free term.

Considering that on the left side of the equation nth polynomial degrees, then the theorem has another interpretation.

Theorem. When dividing a polynomial of nth degree with respect to x by binomial x-a the remainder is equal to the value of the dividend when x = a. (letter a can denote any real or imaginary number, i.e. any complex number) .

Proof: let f(x) denotes an arbitrary polynomial of the nth degree with respect to the variable x and let, when divided by a binomial ( x-a) turned out in private q(x), and the remainder R. It's obvious that q(x) there will be some polynomial (n - 1)th degree relative to x, and the remainder R will be a constant value, i.e. independent of x.

If the remainder R was a polynomial of the first degree with respect to x, then this would mean that the division failed. So, R from x does not depend. By definition of division we obtain the identity: f(x)=(x-a) q(x)+R.

The equality is true for any value of x, which means it is also true for x=a, we get: f(a)=(a-a) q(a)+R. Symbol f(a) denotes the value of the polynomial f (x) at x=a, q(a) stands for value q(x) at x=a. Remainder R remained the same as it was before, because R from x does not depend. Work ( x-a) q(a) = 0, since the factor ( x-a) = 0, and the multiplier q(a) There is certain number. Therefore, from the equality we get: f(a)= R, etc.

Example 1. Find the remainder of a polynomial x 3 - 3x 2 + 6x- 5 per binomial

x- 2. By Bezout’s theorem : R=f(2) = 23-322 + 62 -5=3. Answer: R= 3.

Note that Bezout’s theorem is important not so much in itself as for its consequences. (Annex 1)

Let us dwell on the consideration of some techniques for applying Bezout’s theorem to solving practical problems. It should be noted that when solving equations using Bezout’s theorem, it is necessary:

Find all integer divisors of the free term;

Find at least one root of the equation from these divisors;

Divide the left side of the equation by (Ha);

Write down the product of the divisor and the quotient on the left side of the equation;

Solve the resulting equation.

Let's look at the example of solving the equation x 3 + 4X 2 + x - 6 = 0 .

Solution: find the divisors of the free term ±1 ; ± 2; ± 3; ± 6. Let's calculate the values ​​at x= 1, 1 3 + 41 2 + 1- 6=0. Divide the left side of the equation by ( X- 1). Let’s do the division using a “corner” and get:

Conclusion: Bezout’s theorem is one of the methods that we consider in our work, studied in the elective curriculum. It is difficult to understand, because in order to master it, you need to know all the consequences from it, but at the same time, Bezout’s theorem is one of the main assistants for students on the Unified State Exam.

2.4 Horner scheme

To divide a polynomial by a binomial x-α you can use a special simple technique invented by English mathematicians of the 17th century, later called Horner’s scheme. In addition to finding the roots of equations, using Horner's scheme you can more simply calculate their values. To do this, you need to substitute the value of the variable into the polynomial Pn (x)=a 0 xn+a 1 x n-1 +a 2 xⁿ - ²+…++ a n -1 x+a n. (1)

Consider dividing the polynomial (1) by the binomial x-α.

Let us express the coefficients of the incomplete quotient b 0 xⁿ - ¹+ b 1 xⁿ - ²+ b 2 xⁿ - ³+…+ bn -1 and the remainder r through the coefficients of the polynomial Pn( x) and number α. b 0 =a 0 , b 1 = α b 0 +a 1 , b 2 = α b 1 +a 2 …, bn -1 =

= α bn -2 +a n -1 = α bn -1 +a n .

Calculations using Horner’s scheme are presented in the following table:

	A 0	a 1	a 2 ,
	b 0 =a 0	b 1 = α b 0 +a 1	b 2 = α b 1 +a 2		r=α b n-1 +a n

Because the r=Pn(α), then α is the root of the equation. In order to check whether α is a multiple root, Horner’s scheme can be applied to the quotient b 0 x+ b 1 x+…+ bn -1 according to the table. If in the column under bn -1 the result is 0 again, which means α is a multiple root.

Let's look at an example: solve the equation X 3 + 4X 2 + x - 6 = 0.

Let us apply to the left side of the equation the factorization of the polynomial on the left side of the equation, Horner's scheme.

Solution: find the divisors of the free term ± 1; ± 2; ± 3; ± 6.

				6 ∙ 1 + (-6) = 0

The coefficients of the quotient are the numbers 1, 5, 6, and the remainder r = 0.

Means, X 3 + 4X 2 + X - 6 = (X - 1) (X 2 + 5X + 6) = 0.

From here: X- 1 = 0 or X 2 + 5X + 6 = 0.

X = 1, X 1 = -2; X 2 = -3. Answer: 1,- 2, - 3.

Conclusion: thus, on one equation we have shown the use of two in various ways factorization of polynomials. In our opinion, Horner's scheme is the most practical and economical.

2.5 Solving 4th degree equations. Ferrari method

Cardano's student Ludovic Ferrari discovered a way to solve a fourth-degree equation. The Ferrari method consists of two stages.

Stage I: equations of the form are represented as the product of two square trinomials; this follows from the fact that the equation is of the 3rd degree and has at least one solution.

Stage II: the resulting equations are solved using factorization, but in order to find the required factorization, cubic equations have to be solved.

The idea is to represent the equations in the form A 2 =B 2, where A= x 2 +s,

B-linear function of x. Then it remains to solve the equations A = ±B.

For clarity, consider the equation: Isolating the 4th degree, we get: For any d the expression will be perfect square. Add to both sides of the equation we get

On the left side there is a complete square, you can pick up d, so that the right side of (2) also becomes a complete square. Let's imagine that we have achieved this. Then our equation looks like this:

Finding the root will not be difficult later. To choose the right d it is necessary that the discriminant of the right side of (3) becomes zero, i.e.

So to find d, we need to solve this 3rd degree equation. This auxiliary equation is called resolvent.

We easily find the whole root of the resolvent: d = 1

Substituting the equation into (1) we get

Conclusion: the Ferrari method is universal, but complex and cumbersome. At the same time, if the solution algorithm is clear, then 4th degree equations can be solved using this method.

2.6 Method of uncertain coefficients

The success of solving an equation of the 4th degree using the Ferrari method depends on whether we solve the resolvent - an equation of the 3rd degree, which, as we know, is not always possible.

The essence of the method of indefinite coefficients is that the type of factors into which a given polynomial is decomposed is guessed, and the coefficients of these factors (also polynomials) are determined by multiplying the factors and equating the coefficients at the same powers of the variable.

Example: solve the equation:

Suppose that the left side of our equation can be decomposed into two square trinomials with integer coefficients such that the identical equality is true

Obviously, the coefficients in front of them must be equal to 1, and the free terms must be equal to one + 1, the other - 1.

The coefficients facing the X. Let us denote them by A and and to determine them, we multiply both trinomials on the right side of the equation.

As a result we get:

Equating coefficients at the same degrees X on the left and right sides of equality (1), we obtain a system for finding and

Having solved this system, we will have

So our equation is equivalent to the equation

Having solved it, we get the following roots: .

The method of indefinite coefficients is based on the following statements: any polynomial of the fourth degree in the equation can be decomposed into the product of two polynomials of the second degree; two polynomials are identically equal if and only if their coefficients are equal for the same powers X.

2.7 Symmetric equations

Definition. An equation of the form is called symmetric if the first coefficients on the left of the equation are equal to the first coefficients on the right.

We see that the first coefficients on the left are equal to the first coefficients on the right.

If such an equation has an odd degree, then it has a root X= - 1. Next we can lower the degree of the equation by dividing it by ( x+ 1). It turns out that when dividing a symmetric equation by ( x+ 1) a symmetric equation of even degree is obtained. Proof of the symmetry of the coefficients is presented below. (Appendix 6) Our task is to learn how to solve symmetric equations of even degree.

For example: (1)

Let's solve equation (1), divide by X 2 (to medium degree) = 0.

Let us group terms with symmetric

) + 3(x+ . Let's denote at= x+ , let’s square both sides, hence = at 2 So, 2( at 2 or 2 at 2 + 3 solving the equation, we get at = , at= 3. Next, let's return to replacement x+ = and x+ = 3. We obtain the equations and The first has no solution, and the second has two roots. Answer:.

Conclusion: this type equations are not often encountered, but if you come across one, then it can be solved easily and simply without resorting to cumbersome calculations.

2.8 Isolation of a full degree

Consider the equation.

The left side is the cube of the sum (x+1), i.e.

We extract the third root from both parts: , then we get

Where is the only root?

RESEARCH RESULTS

Based on the results of the work, we came to the following conclusions:

Thanks to the studied theory, we became acquainted with various methods solving entire equations of higher degrees;

D. Cardano's formula is difficult to use and gives a high probability of making errors in the calculation;

− L. Ferrari’s method allows one to reduce the solution to a fourth-degree equation to a cubic one;

− Bezout’s theorem can be used both for cubic equations and for equations of the fourth degree; it is more understandable and visual when applied to solving equations;

Horner's scheme helps to significantly reduce and simplify calculations in solving equations. In addition to finding the roots, using Horner's scheme you can more simply calculate the values ​​of the polynomials on the left side of the equation;

Of particular interest were the solutions of equations by the method of indefinite coefficients and the solution of symmetric equations.

During research work It was found that students become familiar with the simplest methods of solving equations of the highest degree in elective mathematics classes, starting in the 9th or 10th grade, as well as in special courses at visiting mathematics schools. This fact was established as a result of a survey of mathematics teachers of MBOU "Secondary School No. 9" and students who show increased interest to the subject "mathematics".

The most popular methods for solving equations of higher degrees, which are encountered when solving olympiads, competitive problems and as a result of students preparing for exams, are methods based on the application of Bezout’s theorem, Horner’s scheme and the introduction of a new variable.

Demonstration of the results of research work, i.e. methods for solving equations not taught in the school mathematics curriculum interested my classmates.

Conclusion

Having studied educational and scientific literature, Internet resources in youth educational forums