Сумма корней x^3+x^2+x-1=0

Сумма и произведение корней

сумма

           _____________                            /             _____________                       \              _____________                            /           _____________                       \                                                  
          /        ____                             |            /        ____                        |             /        ____                             |          /        ____                        |                                                  
         /  17   \/ 33                              |    ___    /  17   \/ 33                         |            /  17   \/ 33                              |  ___    /  17   \/ 33                         |              _____________                       
      3 /   -- + ------                             |  \/ 3 *3 /   -- + ------             ___        |         3 /   -- + ------                             |\/ 3 *3 /   -- + ------             ___        |             /        ____                        
  1   \/    27     9                1               |        \/    27     9              \/ 3         |     1   \/    27     9                1               |      \/    27     9              \/ 3         |     1      /  17   \/ 33              2          
- - - ------------------ + -------------------- + I*|- ------------------------ - --------------------| + - - - ------------------ + -------------------- + I*|------------------------ + --------------------| + - - + 3 /   -- + ------  - --------------------
  3           2                   _____________     |             2                      _____________|     3           2                   _____________     |           2                      _____________|     3   \/    27     9              _____________
                                 /        ____      |                                   /        ____ |                                    /        ____      |                                 /        ____ |                                    /        ____ 
                                /  17   \/ 33       |                                  /  17   \/ 33  |                                   /  17   \/ 33       |                                /  17   \/ 33  |                                   /  17   \/ 33  
                           9*3 /   -- + ------      |                             9*3 /   -- + ------ |                              9*3 /   -- + ------      |                           9*3 /   -- + ------ |                              9*3 /   -- + ------ 
                             \/    27     9         \                               \/    27     9    /                                \/    27     9         \                             \/    27     9    /                                \/    27     9    

$$\left(- \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)\right)\right)$$

=

       /           _____________                       \     /             _____________                       \
       |          /        ____                        |     |            /        ____                        |
       |  ___    /  17   \/ 33                         |     |    ___    /  17   \/ 33                         |
       |\/ 3 *3 /   -- + ------             ___        |     |  \/ 3 *3 /   -- + ------             ___        |
       |      \/    27     9              \/ 3         |     |        \/    27     9              \/ 3         |
-1 + I*|------------------------ + --------------------| + I*|- ------------------------ - --------------------|
       |           2                      _____________|     |             2                      _____________|
       |                                 /        ____ |     |                                   /        ____ |
       |                                /  17   \/ 33  |     |                                  /  17   \/ 33  |
       |                           9*3 /   -- + ------ |     |                             9*3 /   -- + ------ |
       \                             \/    27     9    /     \                               \/    27     9    /

$$-1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right) + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)$$

произведение

/           _____________                            /             _____________                       \\ /           _____________                            /           _____________                       \\                                                  
|          /        ____                             |            /        ____                        || |          /        ____                             |          /        ____                        ||                                                  
|         /  17   \/ 33                              |    ___    /  17   \/ 33                         || |         /  17   \/ 33                              |  ___    /  17   \/ 33                         || /           _____________                       \
|      3 /   -- + ------                             |  \/ 3 *3 /   -- + ------             ___        || |      3 /   -- + ------                             |\/ 3 *3 /   -- + ------             ___        || |          /        ____                        |
|  1   \/    27     9                1               |        \/    27     9              \/ 3         || |  1   \/    27     9                1               |      \/    27     9              \/ 3         || |  1      /  17   \/ 33              2          |
|- - - ------------------ + -------------------- + I*|- ------------------------ - --------------------||*|- - - ------------------ + -------------------- + I*|------------------------ + --------------------||*|- - + 3 /   -- + ------  - --------------------|
|  3           2                   _____________     |             2                      _____________|| |  3           2                   _____________     |           2                      _____________|| |  3   \/    27     9              _____________|
|                                 /        ____      |                                   /        ____ || |                                 /        ____      |                                 /        ____ || |                                 /        ____ |
|                                /  17   \/ 33       |                                  /  17   \/ 33  || |                                /  17   \/ 33       |                                /  17   \/ 33  || |                                /  17   \/ 33  |
|                           9*3 /   -- + ------      |                             9*3 /   -- + ------ || |                           9*3 /   -- + ------      |                           9*3 /   -- + ------ || |                           9*3 /   -- + ------ |
\                             \/    27     9         \                               \/    27     9    // \                             \/    27     9         \                             \/    27     9    // \                             \/    27     9    /

$$\left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right)\right) \left(- \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}\right)$$

=

1

$$1$$

Теорема Виета

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