Произведение корней x^3-2022*x+sqrt(2021)=0

Сумма и произведение корней

сумма

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                                                                   |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||                                                                      |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||                                               /      ______              ___ 
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                                                                   |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|                                                                      |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|                                           3 /   ----------- + -------------- 
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- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ---------------------------------- + - 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ---------------------------------- + - ----------------------------------- - -----------------------------------
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$$\left(\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) + \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)$$

=

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                                                                                                                                                                      3 /   ----------- + --------------      |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||     |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||              |atan|--------------||
                  2022                         /                         1                         \          /                         1                         \   \/         2              2             |         /                         1                         \              \         3          /               \         3          /|     |         /                         1                         \              \         3          /               \         3          /|     _____    |    \     2021     /|
- ----------------------------------- - 2022*re|---------------------------------------------------| - 2022*re|---------------------------------------------------| - ----------------------------------- + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + \/ 674 *cos|--------------------|
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$$- 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)} - \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right) - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}$$

произведение

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|- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ----------------------------------|*|- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ----------------------------------|*|- ----------------------------------- - -----------------------------------|
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$$\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)$$

=

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    3 ___ |         |      \     2021     /|| |       3 ___ /  ______             ___\   |
337*\/ 2 *|1 - 2*cos|----------------------||*\1348 + \/ 2 *\\/ 2021  + 20205*I*\/ 3 /   /
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$$\frac{337 \sqrt[3]{2} \left(1 - 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}\right) \left(1348 + \sqrt[3]{2} \left(\sqrt{2021} + 20205 \sqrt{3} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{\sqrt{2021} + 20205 \sqrt{3} i}}$$

Теорема Виета

это приведённое кубическое уравнение
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2022$$
$$v = \frac{d}{a}$$
$$v = \sqrt{2021}$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2022$$
$$x_{1} x_{2} x_{3} = \sqrt{2021}$$