Сумма корней cos(x)^3-cos(x)^2+cos(x)=1/3

Сумма и произведение корней

сумма

    /    /                             /          ___\\\              /    /                             /          ___\\\       /    /                             /          ___\\\              /    /                             /          ___\\\                                         /    /                             /          ___\\\     /    /                             /          ___\\\       /    /                             /          ___\\\     /    /                             /          ___\\\                         
    |    |                         2/3 |  1   I*\/ 3 |||              |    |                         2/3 |  1   I*\/ 3 |||       |    |                         2/3 |  1   I*\/ 3 |||              |    |                         2/3 |  1   I*\/ 3 |||                                         |    |                         2/3 |  1   I*\/ 3 |||     |    |                         2/3 |  1   I*\/ 3 |||       |    |                         2/3 |  1   I*\/ 3 |||     |    |                         2/3 |  1   I*\/ 3 |||                         
    |    |          3 ___         2   *|- - + -------|||              |    |          3 ___         2   *|- - + -------|||       |    |          3 ___         2   *|- - - -------|||              |    |          3 ___         2   *|- - - -------|||         /    3 ___    2/3\              |    |          3 ___         2   *|- - + -------|||     |    |          3 ___         2   *|- - + -------|||       |    |          3 ___         2   *|- - - -------|||     |    |          3 ___         2   *|- - - -------|||       /    3 ___    2/3\
    |    |1         \/ 2               \  2      2   /||              |    |1         \/ 2               \  2      2   /||       |    |1         \/ 2               \  2      2   /||              |    |1         \/ 2               \  2      2   /||         |1   \/ 2    2   |              |    |1         \/ 2               \  2      2   /||     |    |1         \/ 2               \  2      2   /||       |    |1         \/ 2               \  2      2   /||     |    |1         \/ 2               \  2      2   /||       |1   \/ 2    2   |
- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|| + - re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|| + - acos|- - ----- + ----| + 2*pi + I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|| + I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|| + acos|- - ----- + ----|
    |    |3     /          ___\            3          ||              |    |3     /          ___\            3          ||       |    |3     /          ___\            3          ||              |    |3     /          ___\            3          ||         \3     3      3  /              |    |3     /          ___\            3          ||     |    |3     /          ___\            3          ||       |    |3     /          ___\            3          ||     |    |3     /          ___\            3          ||       \3     3      3  /
    |    |      |  1   I*\/ 3 |                       ||              |    |      |  1   I*\/ 3 |                       ||       |    |      |  1   I*\/ 3 |                       ||              |    |      |  1   I*\/ 3 |                       ||                                         |    |      |  1   I*\/ 3 |                       ||     |    |      |  1   I*\/ 3 |                       ||       |    |      |  1   I*\/ 3 |                       ||     |    |      |  1   I*\/ 3 |                       ||                         
    |    |    3*|- - + -------|                       ||              |    |    3*|- - + -------|                       ||       |    |    3*|- - - -------|                       ||              |    |    3*|- - - -------|                       ||                                         |    |    3*|- - + -------|                       ||     |    |    3*|- - + -------|                       ||       |    |    3*|- - - -------|                       ||     |    |    3*|- - - -------|                       ||                         
    \    \      \  2      2   /                       //              \    \      \  2      2   /                       //       \    \      \  2      2   /                       //              \    \      \  2      2   /                       //                                         \    \      \  2      2   /                       //     \    \      \  2      2   /                       //       \    \      \  2      2   /                       //     \    \      \  2      2   /                       //                         

$$\operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + \left(\left(\left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) + \left(\left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) + \left(\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right)\right)\right)\right) + \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right)\right)$$

=

6*pi

$$6 \pi$$

произведение

/    /    /                             /          ___\\\              /    /                             /          ___\\\\ /    /    /                             /          ___\\\              /    /                             /          ___\\\\                                   /    /    /                             /          ___\\\     /    /                             /          ___\\\\ /    /    /                             /          ___\\\     /    /                             /          ___\\\\                       
|    |    |                         2/3 |  1   I*\/ 3 |||              |    |                         2/3 |  1   I*\/ 3 |||| |    |    |                         2/3 |  1   I*\/ 3 |||              |    |                         2/3 |  1   I*\/ 3 ||||                                   |    |    |                         2/3 |  1   I*\/ 3 |||     |    |                         2/3 |  1   I*\/ 3 |||| |    |    |                         2/3 |  1   I*\/ 3 |||     |    |                         2/3 |  1   I*\/ 3 ||||                       
|    |    |          3 ___         2   *|- - + -------|||              |    |          3 ___         2   *|- - + -------|||| |    |    |          3 ___         2   *|- - - -------|||              |    |          3 ___         2   *|- - - -------|||| /      /    3 ___    2/3\       \ |    |    |          3 ___         2   *|- - + -------|||     |    |          3 ___         2   *|- - + -------|||| |    |    |          3 ___         2   *|- - - -------|||     |    |          3 ___         2   *|- - - -------||||     /    3 ___    2/3\
|    |    |1         \/ 2               \  2      2   /||              |    |1         \/ 2               \  2      2   /||| |    |    |1         \/ 2               \  2      2   /||              |    |1         \/ 2               \  2      2   /||| |      |1   \/ 2    2   |       | |    |    |1         \/ 2               \  2      2   /||     |    |1         \/ 2               \  2      2   /||| |    |    |1         \/ 2               \  2      2   /||     |    |1         \/ 2               \  2      2   /|||     |1   \/ 2    2   |
|- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|||*|- re|acos|- - ----------------- + --------------------|| + 2*pi - I*im|acos|- - ----------------- + --------------------|||*|- acos|- - ----- + ----| + 2*pi|*|I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|||*|I*im|acos|- - ----------------- + --------------------|| + re|acos|- - ----------------- + --------------------|||*acos|- - ----- + ----|
|    |    |3     /          ___\            3          ||              |    |3     /          ___\            3          ||| |    |    |3     /          ___\            3          ||              |    |3     /          ___\            3          ||| \      \3     3      3  /       / |    |    |3     /          ___\            3          ||     |    |3     /          ___\            3          ||| |    |    |3     /          ___\            3          ||     |    |3     /          ___\            3          |||     \3     3      3  /
|    |    |      |  1   I*\/ 3 |                       ||              |    |      |  1   I*\/ 3 |                       ||| |    |    |      |  1   I*\/ 3 |                       ||              |    |      |  1   I*\/ 3 |                       |||                                   |    |    |      |  1   I*\/ 3 |                       ||     |    |      |  1   I*\/ 3 |                       ||| |    |    |      |  1   I*\/ 3 |                       ||     |    |      |  1   I*\/ 3 |                       |||                       
|    |    |    3*|- - + -------|                       ||              |    |    3*|- - + -------|                       ||| |    |    |    3*|- - - -------|                       ||              |    |    3*|- - - -------|                       |||                                   |    |    |    3*|- - + -------|                       ||     |    |    3*|- - + -------|                       ||| |    |    |    3*|- - - -------|                       ||     |    |    3*|- - - -------|                       |||                       
\    \    \      \  2      2   /                       //              \    \      \  2      2   /                       /// \    \    \      \  2      2   /                       //              \    \      \  2      2   /                       ///                                   \    \    \      \  2      2   /                       //     \    \      \  2      2   /                       /// \    \    \      \  2      2   /                       //     \    \      \  2      2   /                       ///                       

$$\left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) \left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)}{3} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(\frac{1}{3} + \frac{2^{\frac{2}{3}} \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)}{3} - \frac{\sqrt[3]{2}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)} \right)}\right)}\right) \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$

=

/      /    3 ___    2/3\       \ /    /    /     2/3   3 ___     3 ___   ___      2/3   ___\\     /    /     2/3   3 ___     3 ___   ___      2/3   ___\\\ /    /    /     2/3   3 ___     3 ___   ___      2/3   ___\\     /    /     2/3   3 ___     3 ___   ___      2/3   ___\\\ /            /    /     2/3   3 ___     3 ___   ___      2/3   ___\\     /    /     2/3   3 ___     3 ___   ___      2/3   ___\\\ /            /    /     2/3   3 ___     3 ___   ___      2/3   ___\\     /    /     2/3   3 ___     3 ___   ___      2/3   ___\\\     /    3 ___    2/3\
|      |1   \/ 2    2   |       | |    |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||     |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||| |    |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||     |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||| |            |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||     |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||| |            |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 ||     |    |1   2      \/ 2    I*\/ 2 *\/ 3    I*2   *\/ 3 |||     |1   \/ 2    2   |
|- acos|- - ----- + ----| + 2*pi|*|I*im|acos|- - ---- + ----- - ------------- - ------------|| + re|acos|- - ---- + ----- - ------------- - ------------|||*|I*im|acos|- - ---- + ----- + ------------- + ------------|| + re|acos|- - ---- + ----- + ------------- + ------------|||*|-2*pi + I*im|acos|- - ---- + ----- - ------------- - ------------|| + re|acos|- - ---- + ----- - ------------- - ------------|||*|-2*pi + I*im|acos|- - ---- + ----- + ------------- + ------------|| + re|acos|- - ---- + ----- + ------------- + ------------|||*acos|- - ----- + ----|
\      \3     3      3  /       / \    \    \3    6       6           6              6      //     \    \3    6       6           6              6      /// \    \    \3    6       6           6              6      //     \    \3    6       6           6              6      /// \            \    \3    6       6           6              6      //     \    \3    6       6           6              6      /// \            \    \3    6       6           6              6      //     \    \3    6       6           6              6      ///     \3     3      3  /

$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)}\right) \left(- \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)} + 2 \pi\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} + \frac{\sqrt[3]{2} \sqrt{3} i}{6} + \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{2^{\frac{2}{3}}}{6} + \frac{\sqrt[3]{2}}{6} + \frac{1}{3} - \frac{2^{\frac{2}{3}} \sqrt{3} i}{6} - \frac{\sqrt[3]{2} \sqrt{3} i}{6} \right)}\right)}\right) \operatorname{acos}{\left(- \frac{\sqrt[3]{2}}{3} + \frac{1}{3} + \frac{2^{\frac{2}{3}}}{3} \right)}$$