Example 2. Let's find the sum of the cubes of the roots of the equation

$$
x^{3}+2 x^{2}+x-3=0
$$
S o l u t i o n. This can be done in various ways, for example, by sequentially calculating the sum of the squares of the roots and then the sum of the cubes of the roots. However, we will use a frequently applied identity in mathematics:

$$
\begin{aligned}
a^{3}+b^{3}+c^{3}-3 a b c= & (a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-\right. \\
& -a c-b c) .
\end{aligned}
$$

In our case, this identity will take the form:

$$
\begin{gathered}
x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=3 x_{1} x_{2} x_{3}+\left(x_{1}+x_{2}+x_{3}\right)\left(x_{1}^{2}+x_{2}^{2}+\right. \\
\left.+x_{3}^{2}-x_{1} x_{2}-x_{2} x_{3}-x_{3} x_{1}\right)
\end{gathered}
$$

We will express \( S_{3}=x_{1}^{3}+x_{2}^{3}+x_{3}^{3} \) in terms of \( \sigma_{1}, \sigma_{2} \) and \( \sigma_{3} \), where \( \sigma_{1}=x_{1}+x_{2}+x_{3}, \sigma_{2}=x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}, \sigma_{3}=x_{1} x_{2} x_{3} \).

The sum \( x_{1}^{2}+x_{2}^{2}+x_{3}^{2} \) can be easily expressed in terms of \( \sigma_{1} \) and \( \sigma_{2} \):

$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=\sigma_{1}^{2}-2 \sigma_{2}
$$

Finally, we get:

$$
x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=\sigma_{1}^{3}-3 \sigma_{1} \sigma_{2}+3 \sigma_{3} .
$$

Since by the condition (by Vieta's formulas) \( \sigma_{1}=-2 \),

$$
\sigma_{2}=1, \sigma_{3}=3, \text { then } x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=7
$$

E x a m p l e. Solve the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=2 \\
x^{2}+y^{2}+z^{2}=6 \\
x^{3}+y^{3}+z^{3}=8
\end{array}\right.
$$

S o l u t i o n. Consider the triplet of variables \( x, y, z \) as the roots of some cubic equation \( t^{3}+a t^{2}+b t+c=0 \) and find the coefficients \( a, b, c \).

By Vieta's formulas

$$
\sigma_{1}=-a, \quad \sigma_{2}=b, \quad \sigma_{3}=-c
$$

But from the previous example,

$$
\begin{aligned}
x^{2}+y^{2}+z^{2}=\sigma_{1}^{2}-2 \sigma_{2} & =a^{2}-2 b \\
x^{3}+y^{3}+z^{3}=\sigma_{1}^{3}-3 \sigma_{1} \sigma_{2}+3 \sigma_{3} & =-a^{3}+3 a b-3 c
\end{aligned}
$$

Substituting these expressions into the original system of equations, we get:

$$
\left\{\begin{array}{l}
-a=2 \\
a^{2}-2 b=6 \\
-a^{3}+3 a b-3 c=8
\end{array}\right.
$$

from which we easily find: \( a=-2, b=-1, c=2 \).

Therefore, the desired equation will be:

$$
t^{3}-2 t^{2}-t+2=0
$$

or

$$
\left(t^{2}-1\right)(t-2)=0
$$

Its roots are: \( -1, 1, 2 \). Since the original system is symmetric, any combination of this triplet of numbers is its solution.

## EXERCISES

11. Formulate an equation whose roots are the lengths of the radii \( r_{a}, r_{b}, r_{c} \) of the excircles.

12*. Formulate a cubic equation whose roots are

$$
\cos \frac{\pi}{7}, \cos \frac{3 \pi}{7}, \cos \frac{5 \pi}{7}
$$

13*. Find the sum of the 16th powers of the roots of the equation

$$
x^{3}-x+1=0 \text {. }
$$

14. Calculate the area of a quadrilateral whose side lengths are the roots of the equation \( x^{4}+k x^{3}+m x^{2}+n x+5=0 \), and the sum of the measures of the opposite angles is \( \pi \).
15. If the equation \( a x^{3}-3 b x^{2}+3 c x-d=0 \) has roots \( \frac{b}{a} \), \( \frac{c}{b} \), \( \frac{d}{c} \), then these roots are either equal or form an arithmetic progression. Prove it.
16. Calculate the values of the expressions: \( x_{1}^{2} x_{2}^{2}+x_{2}^{2} x_{3}^{2}+x_{3}^{2} x_{1}^{2} \); \( x_{1}^{2} x_{2}+x_{1} x_{2}^{2}+x_{2}^{2} x_{3}+x_{2} x_{3}^{2}+x_{1}^{2} x_{3}+x_{1} x_{3}^{2} \), if \( x_{1}, x_{2}, x_{3} \) are the roots of the equation \( x^{3}-x+3=0 \).
17. Find the real roots of the systems of equations:
a) \( \left\{\begin{aligned} x+y+z & =4 \\ x y+y z+z x & =5 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} & =\frac{5}{2}\end{aligned}\right. \)
b) \( \left\{\begin{aligned} x^{3}+y^{3}+z^{3} & =\frac{73}{8}, \\ x y+x z+y z & =x+y+z \\ x y z & =1 .\end{aligned}\right. \)
18. Given the equation

$$
x^{3}+p x^{2}+q x+z=0
$$

Find the relationship between the coefficients under which one root is equal to the sum of the other two.

19. Given the equation \( x^{4}+a x^{3}+b x^{2}+c x+d=0 \), in which: a) \( x_{1}+x_{2}=x_{3}+x_{4} \); b) \( x_{1} x_{2}=x_{3} x_{4} \). Find the relationship between the coefficients. Find ways to solve the equation under these conditions.
20. Determine the condition that the coefficients of the equation \( x^{3}+p x^{2}+q x+r=0 \) must satisfy for its three roots to form an arithmetic progression.
21. What necessary and sufficient condition must the coefficients of the equation \( x^{8}+p x^{2}+q x+r=0 \), where \( r \neq 0 \), \( q \neq 0 \), satisfy if its roots are related by the equation \( \frac{1}{x_{1}}+\frac{1}{x_{2}}=\frac{1}{x_{3}} \) ?
22. Determine \( \lambda \) so that one of the roots of the equation \( x^{3}+7 x+\lambda=0 \) is twice the second.

12

Problems on forming equations of the 3rd and 4th degrees. The solutions to the geometric problems presented in this section mainly reduce to solving reciprocal or other equations, which in turn can be appropriately reduced to quadratic equations.

Let's consider this on a specific example.