Сумма и произведение корней
[src] __________________ / __________________ \ __________________ / __________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 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 /
$$5 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} + \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}\right) + i \left(- \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2}\right)$$
/ __________________ / __________________ \\ / __________________ / __________________ \\
| / ______ | / ______ || | / ______ | / ______ ||
| / 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 /
$$\left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} - \frac{37}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}{2} + \frac{37 \sqrt{3}}{18 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}}\right)\right) \left(\frac{37}{9 \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}} + \frac{5}{3} + \sqrt[3]{\frac{2 \sqrt{5127}}{9} + \frac{485}{27}}\right)$$
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
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 5$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4$$
$$x_{1} x_{2} x_{3} = -20$$