Решение уравнений/ Любые/ x^4+x^3+2*x^2+2*x+3=0

Произведение корней x^4+x^3+2*x^2+2*x+3=0

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    Решение

    Сумма и произведение корней [src]
    сумма
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                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17                                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17                                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17                                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17             
                                                                                     I*           /    -- + 2*3 /   - --- + ----------  - ------------------------------------------------------------------------------ + ---------------------------                                                                                         I*           /    -- + 2*3 /   - --- + ----------  + ------------------------------------------------------------------------------ + ---------------------------                                                                                         I*           /    -- + 2*3 /   - --- + ----------  + ------------------------------------------------------------------------------ + ---------------------------                                                                                         I*           /    -- + 2*3 /   - --- + ----------  - ------------------------------------------------------------------------------ + --------------------------- 
                __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________                    __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________                    __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________                    __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________ 
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              /               /             ______                                             /                                                    /               /             ______                                        /    263   I*\/ 1087                    /               /             ______                                             /                                                    /               /             ______                                        /    263   I*\/ 1087                    /               /             ______                                             /                                                    /               /             ______                                        /    263   I*\/ 1087                    /               /             ______                                             /                                                    /               /             ______                                        /    263   I*\/ 1087   
             /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------                  /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------                  /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------                  /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------  
            /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48                     /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48                     /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48                     /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48      
           /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                              /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                              /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                              /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                
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         /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                              /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                              /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                              /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                  
        /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                            /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                            /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                            /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                 
  1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                          1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                          1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                          1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                     
- - - ---------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------ + - - + ---------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------ + - - + ---------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------ + - - - ---------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------
  4                                        2                                                                                                                         2                                                                                      4                                        2                                                                                                                         2                                                                                      4                                        2                                                                                                                         2                                                                                      4                                        2                                                                                                                         2
    $$\left(\left(\left(- \frac{1}{4} - \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right) + \left(- \frac{1}{4} - \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} - \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} - \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right)\right) + \left(- \frac{1}{4} + \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right)\right) + \left(- \frac{1}{4} - \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} - \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right)$$
    =
    -1
    $$-1$$
    произведение
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|                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17             | |                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17             | |                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17             | |                                                                                                   /   13        /    263   I*\/ 1087                                           9                                                       17             |
|                                                                                     I*           /    -- + 2*3 /   - --- + ----------  - ------------------------------------------------------------------------------ + --------------------------- | |                                                                                     I*           /    -- + 2*3 /   - --- + ----------  + ------------------------------------------------------------------------------ + --------------------------- | |                                                                                     I*           /    -- + 2*3 /   - --- + ----------  + ------------------------------------------------------------------------------ + --------------------------- | |                                                                                     I*           /    -- + 2*3 /   - --- + ----------  - ------------------------------------------------------------------------------ + --------------------------- |
|                __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________ | |                __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________ | |                __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________ | |                __________________________________________________________________               /     6      \/      432       48                    __________________________________________________________________          ____________________ |
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|             /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------  | |             /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------  | |             /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------  | |             /     13        /    263   I*\/ 1087                  17                         /                                                    /     13        /    263   I*\/ 1087                  17                9*3 /   - --- + ----------  |
|            /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48      | |            /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48      | |            /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48      | |            /    - -- + 2*3 /   - --- + ----------  + ---------------------------            /                                            4*      /    - -- + 2*3 /   - --- + ----------  + ---------------------------      \/      432       48      |
|           /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                | |           /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                | |           /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                | |           /       12     \/      432       48               ____________________           /                                                    /       12     \/      432       48               ____________________                                |
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|         /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                  | |         /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                  | |         /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                  | |         /                                                 /    263   I*\/ 1087           /                                                    /                                                 /    263   I*\/ 1087                                  |
|        /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                 | |        /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                 | |        /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                 | |        /                                             9*3 /   - --- + ----------         /                                                    /                                             9*3 /   - --- + ----------                                 |
|  1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                     | |  1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                     | |  1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                     | |  1   \/                                                \/      432       48           \/                                                   \/                                                \/      432       48                                     |
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\  4                                        2                                                                                                                         2                                                                                 / \  4                                        2                                                                                                                         2                                                                                 / \  4                                        2                                                                                                                         2                                                                                 / \  4                                        2                                                                                                                         2                                                                                 /
    $$\left(- \frac{1}{4} - \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} - \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} - \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right) \left(- \frac{1}{4} - \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right) \left(- \frac{1}{4} + \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right) \left(- \frac{1}{4} - \frac{\sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2} + \frac{i \sqrt{\frac{13}{6} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} - \frac{9}{4 \sqrt{- \frac{13}{12} + \frac{17}{9 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}} + 2 \sqrt[3]{- \frac{263}{432} + \frac{\sqrt{1087} i}{48}}}}{2}\right)$$
    =
    3
    $$3$$