Произведение корней x^3-5*x^2-4*x-20=0

Сумма и произведение корней

сумма

                                      __________________     /             __________________                             \                                         __________________     /           __________________                             \                                                          
                                     /           ______      |            /           ______                              |                                        /           ______      |          /           ______                              |                                                          
                                    /  485   2*\/ 5127       |    ___    /  485   2*\/ 5127                               |                                       /  485   2*\/ 5127       |  ___    /  485   2*\/ 5127                               |            __________________                            
                                 3 /   --- + ----------      |  \/ 3 *3 /   --- + ----------                  ___         |                                    3 /   --- + ----------      |\/ 3 *3 /   --- + ----------                  ___         |           /           ______                             
5               37               \/     27       9           |        \/     27       9                  37*\/ 3          |   5               37               \/     27       9           |      \/     27       9                  37*\/ 3          |   5      /  485   2*\/ 5127                 37           
- - -------------------------- - ----------------------- + I*|- ----------------------------- + --------------------------| + - - -------------------------- - ----------------------- + I*|----------------------------- - --------------------------| + - + 3 /   --- + ----------  + -------------------------
3           __________________              2                |                2                         __________________|   3           __________________              2                |              2                         __________________|   3   \/     27       9                __________________
           /           ______                                |                                         /           ______ |              /           ______                                |                                       /           ______ |                                       /           ______ 
          /  485   2*\/ 5127                                 |                                        /  485   2*\/ 5127  |             /  485   2*\/ 5127                                 |                                      /  485   2*\/ 5127  |                                      /  485   2*\/ 5127  
    18*3 /   --- + ----------                                |                                  18*3 /   --- + ---------- |       18*3 /   --- + ----------                                |                                18*3 /   --- + ---------- |                                 9*3 /   --- + ---------- 
       \/     27       9                                     \                                     \/     27       9      /          \/     27       9                                     \                                   \/     27       9      /                                   \/     27       9      

$$\left(\frac{37}{9 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} + \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2}\right)\right)\right)$$

=

      /           __________________                             \     /             __________________                             \
      |          /           ______                              |     |            /           ______                              |
      |  ___    /  485   2*\/ 5127                               |     |    ___    /  485   2*\/ 5127                               |
      |\/ 3 *3 /   --- + ----------                  ___         |     |  \/ 3 *3 /   --- + ----------                  ___         |
      |      \/     27       9                  37*\/ 3          |     |        \/     27       9                  37*\/ 3          |
5 + I*|----------------------------- - --------------------------| + I*|- ----------------------------- + --------------------------|
      |              2                         __________________|     |                2                         __________________|
      |                                       /           ______ |     |                                         /           ______ |
      |                                      /  485   2*\/ 5127  |     |                                        /  485   2*\/ 5127  |
      |                                18*3 /   --- + ---------- |     |                                  18*3 /   --- + ---------- |
      \                                   \/     27       9      /     \                                     \/     27       9      /

$$5 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} + \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}\right) + i \left(- \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2}\right)$$

произведение

/                                      __________________     /             __________________                             \\ /                                      __________________     /           __________________                             \\                                                          
|                                     /           ______      |            /           ______                              || |                                     /           ______      |          /           ______                              ||                                                          
|                                    /  485   2*\/ 5127       |    ___    /  485   2*\/ 5127                               || |                                    /  485   2*\/ 5127       |  ___    /  485   2*\/ 5127                               || /         __________________                            \
|                                 3 /   --- + ----------      |  \/ 3 *3 /   --- + ----------                  ___         || |                                 3 /   --- + ----------      |\/ 3 *3 /   --- + ----------                  ___         || |        /           ______                             |
|5               37               \/     27       9           |        \/     27       9                  37*\/ 3          || |5               37               \/     27       9           |      \/     27       9                  37*\/ 3          || |5      /  485   2*\/ 5127                 37           |
|- - -------------------------- - ----------------------- + I*|- ----------------------------- + --------------------------||*|- - -------------------------- - ----------------------- + I*|----------------------------- - --------------------------||*|- + 3 /   --- + ----------  + -------------------------|
|3           __________________              2                |                2                         __________________|| |3           __________________              2                |              2                         __________________|| |3   \/     27       9                __________________|
|           /           ______                                |                                         /           ______ || |           /           ______                                |                                       /           ______ || |                                    /           ______ |
|          /  485   2*\/ 5127                                 |                                        /  485   2*\/ 5127  || |          /  485   2*\/ 5127                                 |                                      /  485   2*\/ 5127  || |                                   /  485   2*\/ 5127  |
|    18*3 /   --- + ----------                                |                                  18*3 /   --- + ---------- || |    18*3 /   --- + ----------                                |                                18*3 /   --- + ---------- || |                              9*3 /   --- + ---------- |
\       \/     27       9                                     \                                     \/     27       9      // \       \/     27       9                                     \                                   \/     27       9      // \                                \/     27       9      /

$$\left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} + \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}\right)\right) \left(\frac{37}{9 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}\right)$$

=

20

$$20$$

Теорема Виета

это приведённое кубическое уравнение
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = -4$$
$$v = \frac{d}{a}$$
$$v = -20$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4$$
$$x_{1} x_{2} x_{3} = -20$$