Сумма корней x^3-2*x^2-x-2=0

Сумма и произведение корней

сумма

                                  ______________     /             ______________                         \                                     ______________     /           ______________                         \                                                  
                                 /        _____      |            /        _____                          |                                    /        _____      |          /        _____                          |                                                  
                                /  44   \/ 177       |    ___    /  44   \/ 177                           |                                   /  44   \/ 177       |  ___    /  44   \/ 177                           |            ______________                        
                             3 /   -- + -------      |  \/ 3 *3 /   -- + -------               ___        |                                3 /   -- + -------      |\/ 3 *3 /   -- + -------               ___        |           /        _____                         
2             7              \/    27      9         |        \/    27      9              7*\/ 3         |   2             7              \/    27      9         |      \/    27      9              7*\/ 3         |   2      /  44   \/ 177               7          
- - ---------------------- - ------------------- + I*|- ------------------------- + ----------------------| + - - ---------------------- - ------------------- + I*|------------------------- - ----------------------| + - + 3 /   -- + -------  + ---------------------
3           ______________            2              |              2                       ______________|   3           ______________            2              |            2                       ______________|   3   \/    27      9              ______________
           /        _____                            |                                     /        _____ |              /        _____                            |                                   /        _____ |                                   /        _____ 
          /  44   \/ 177                             |                                    /  44   \/ 177  |             /  44   \/ 177                             |                                  /  44   \/ 177  |                                  /  44   \/ 177  
    18*3 /   -- + -------                            |                              18*3 /   -- + ------- |       18*3 /   -- + -------                            |                            18*3 /   -- + ------- |                             9*3 /   -- + ------- 
       \/    27      9                               \                                 \/    27      9    /          \/    27      9                               \                               \/    27      9    /                               \/    27      9    

$$\left(\frac{7}{9 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} + \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + i \left(- \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2}\right)\right)\right)$$

=

      /           ______________                         \     /             ______________                         \
      |          /        _____                          |     |            /        _____                          |
      |  ___    /  44   \/ 177                           |     |    ___    /  44   \/ 177                           |
      |\/ 3 *3 /   -- + -------               ___        |     |  \/ 3 *3 /   -- + -------               ___        |
      |      \/    27      9              7*\/ 3         |     |        \/    27      9              7*\/ 3         |
2 + I*|------------------------- - ----------------------| + I*|- ------------------------- + ----------------------|
      |            2                       ______________|     |              2                       ______________|
      |                                   /        _____ |     |                                     /        _____ |
      |                                  /  44   \/ 177  |     |                                    /  44   \/ 177  |
      |                            18*3 /   -- + ------- |     |                              18*3 /   -- + ------- |
      \                               \/    27      9    /     \                                 \/    27      9    /

$$2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} + \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}\right) + i \left(- \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2}\right)$$

произведение

/                                  ______________     /             ______________                         \\ /                                  ______________     /           ______________                         \\                                                  
|                                 /        _____      |            /        _____                          || |                                 /        _____      |          /        _____                          ||                                                  
|                                /  44   \/ 177       |    ___    /  44   \/ 177                           || |                                /  44   \/ 177       |  ___    /  44   \/ 177                           || /         ______________                        \
|                             3 /   -- + -------      |  \/ 3 *3 /   -- + -------               ___        || |                             3 /   -- + -------      |\/ 3 *3 /   -- + -------               ___        || |        /        _____                         |
|2             7              \/    27      9         |        \/    27      9              7*\/ 3         || |2             7              \/    27      9         |      \/    27      9              7*\/ 3         || |2      /  44   \/ 177               7          |
|- - ---------------------- - ------------------- + I*|- ------------------------- + ----------------------||*|- - ---------------------- - ------------------- + I*|------------------------- - ----------------------||*|- + 3 /   -- + -------  + ---------------------|
|3           ______________            2              |              2                       ______________|| |3           ______________            2              |            2                       ______________|| |3   \/    27      9              ______________|
|           /        _____                            |                                     /        _____ || |           /        _____                            |                                   /        _____ || |                                /        _____ |
|          /  44   \/ 177                             |                                    /  44   \/ 177  || |          /  44   \/ 177                             |                                  /  44   \/ 177  || |                               /  44   \/ 177  |
|    18*3 /   -- + -------                            |                              18*3 /   -- + ------- || |    18*3 /   -- + -------                            |                            18*3 /   -- + ------- || |                          9*3 /   -- + ------- |
\       \/    27      9                               \                                 \/    27      9    // \       \/    27      9                               \                               \/    27      9    // \                            \/    27      9    /

$$\left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + i \left(- \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} - \frac{7}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}{2} + \frac{7 \sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}}\right)\right) \left(\frac{7}{9 \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}} + \frac{2}{3} + \sqrt[3]{\frac{\sqrt{177}}{9} + \frac{44}{27}}\right)$$

=

2

$$2$$

Теорема Виета

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