Произведение корней (x+1)^4+(x+2)^4=(x+3)^4
Решение
Сумма и произведение корней
[src]
сумма
_______________ / _______________ \ _______________ / _______________ \ _______________
/ _____ | / _____ ___ | / _____ | / _____ ___ | / _____
2 / 155 \/ 849 16 | ___ / 155 \/ 849 16*\/ 3 | 2 / 155 \/ 849 16 | ___ / 155 \/ 849 16*\/ 3 | 2 / 155 \/ 849 32
-2 + - - 3 / --- + ------- - ---------------------- + I*|- \/ 3 *3 / --- + ------- + ----------------------| + - - 3 / --- + ------- - ---------------------- + I*|\/ 3 *3 / --- + ------- - ----------------------| + - + 2*3 / --- + ------- + ----------------------
3 \/ 54 18 _______________ | \/ 54 18 _______________| 3 \/ 54 18 _______________ | \/ 54 18 _______________| 3 \/ 54 18 _______________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 155 \/ 849 | / 155 \/ 849 | / 155 \/ 849 | / 155 \/ 849 | / 155 \/ 849
9*3 / --- + ------- | 9*3 / --- + ------- | 9*3 / --- + ------- | 9*3 / --- + ------- | 9*3 / --- + -------
\/ 54 18 \ \/ 54 18 / \/ 54 18 \ \/ 54 18 / \/ 54 18
$$\left(\frac{2}{3} + \frac{32}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + 2 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}\right) + \left(\left(-2 + \left(- \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} - \frac{16}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \frac{2}{3} + i \left(- \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} + \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}}\right)\right)\right) + \left(- \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} - \frac{16}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \frac{2}{3} + i \left(- \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}\right)\right)\right)$$
=
/ _______________ \ / _______________ \ | / _____ ___ | | / _____ ___ | | ___ / 155 \/ 849 16*\/ 3 | | ___ / 155 \/ 849 16*\/ 3 | I*|\/ 3 *3 / --- + ------- - ----------------------| + I*|- \/ 3 *3 / --- + ------- + ----------------------| | \/ 54 18 _______________| | \/ 54 18 _______________| | / _____ | | / _____ | | / 155 \/ 849 | | / 155 \/ 849 | | 9*3 / --- + ------- | | 9*3 / --- + ------- | \ \/ 54 18 / \ \/ 54 18 /
$$i \left(- \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} + \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}}\right) + i \left(- \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}\right)$$
произведение
/ _______________ / _______________ \\ / _______________ / _______________ \\ / _______________ \ | / _____ | / _____ ___ || | / _____ | / _____ ___ || | / _____ | |2 / 155 \/ 849 16 | ___ / 155 \/ 849 16*\/ 3 || |2 / 155 \/ 849 16 | ___ / 155 \/ 849 16*\/ 3 || |2 / 155 \/ 849 32 | -2*|- - 3 / --- + ------- - ---------------------- + I*|- \/ 3 *3 / --- + ------- + ----------------------||*|- - 3 / --- + ------- - ---------------------- + I*|\/ 3 *3 / --- + ------- - ----------------------||*|- + 2*3 / --- + ------- + ----------------------| |3 \/ 54 18 _______________ | \/ 54 18 _______________|| |3 \/ 54 18 _______________ | \/ 54 18 _______________|| |3 \/ 54 18 _______________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 155 \/ 849 | / 155 \/ 849 || | / 155 \/ 849 | / 155 \/ 849 || | / 155 \/ 849 | | 9*3 / --- + ------- | 9*3 / --- + ------- || | 9*3 / --- + ------- | 9*3 / --- + ------- || | 9*3 / --- + ------- | \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 /
$$\left(- \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} - \frac{16}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \frac{2}{3} + i \left(- \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}\right)\right) \left(- 2 \left(- \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} - \frac{16}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + \frac{2}{3} + i \left(- \sqrt{3} \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}} + \frac{16 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}}\right)\right)\right) \left(\frac{2}{3} + \frac{32}{9 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}} + 2 \sqrt[3]{\frac{\sqrt{849}}{18} + \frac{155}{54}}\right)$$
=
-64
$$-64$$