x^7+128=0 (уравнение)

Найду корень уравнения: x^7+128=0

Вы ввели

 7          
x  + 128 = 0

$$x^{7} + 128 = 0$$

Подробное решение

Дано уравнение
$$x^{7} + 128 = 0$$
Т.к. степень в ур-нии равна = 7 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень 7-й степени из обеих частей ур-ния:
Получим:
$$\sqrt[7]{x^{7}} = \sqrt[7]{-128}$$
или
$$x = 2 \sqrt[7]{-1}$$
Раскрываем скобочки в правой части ур-ния

x = -2*1^1/7

Получим ответ: x = 2*(-1)^(1/7)

Остальные 6 корня(ей) являются комплексными.
сделаем замену:
$$z = x$$
тогда ур-ние будет таким:
$$z^{7} = -128$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{7} e^{7 i p} = -128$$
где
$$r = 2$$
- модуль комплексного числа
Подставляем r:
$$e^{7 i p} = -1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(7 p \right)} + \cos{\left(7 p \right)} = -1$$
значит
$$\cos{\left(7 p \right)} = -1$$
и
$$\sin{\left(7 p \right)} = 0$$
тогда
$$p = \frac{2 \pi N}{7} + \frac{\pi}{7}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для z
Значит, решением будет для z:
$$z_{1} = 2 \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)}$$
$$z_{2} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \sin^{2}{\left(\frac{\pi}{7} \right)}$$
$$z_{3} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} + 2 \sin^{2}{\left(\frac{\pi}{7} \right)} - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$z_{4} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$z_{5} = 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} - 2 i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}$$
$$z_{6} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$z_{7} = - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = 2 \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)}$$
$$x_{2} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \sin^{2}{\left(\frac{\pi}{7} \right)}$$
$$x_{3} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} + 2 \sin^{2}{\left(\frac{\pi}{7} \right)} - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$x_{4} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$x_{5} = 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} - 2 i \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}$$
$$x_{6} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$
$$x_{7} = - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 i \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}$$

Быстрый ответ

            2/pi\        2/pi\
x1 = - 2*cos |--| - 2*sin |--|
             \7 /         \7 /

$$x_{1} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \sin^{2}{\left(\frac{\pi}{7} \right)}$$

       /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\
x2 = I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----|
       \     \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /

$$x_{2} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

       /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\
x3 = I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----|
       \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /

$$x_{3} = 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)$$

       /       /3*pi\    /pi\        /pi\    /3*pi\\        /pi\    /3*pi\        /pi\    /3*pi\
x4 = I*|- 2*cos|----|*sin|--| + 2*cos|--|*sin|----|| - 2*cos|--|*cos|----| - 2*sin|--|*sin|----|
       \       \ 7  /    \7 /        \7 /    \ 7  //        \7 /    \ 7  /        \7 /    \ 7  /

$$x_{4} = - 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

       /       /pi\    /3*pi\        /3*pi\    /pi\\        /pi\    /3*pi\        /pi\    /3*pi\
x5 = I*|- 2*cos|--|*sin|----| - 2*cos|----|*sin|--|| - 2*cos|--|*cos|----| + 2*sin|--|*sin|----|
       \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /

$$x_{5} = - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)$$

          /pi\          /pi\
x6 = 2*cos|--| + 2*I*sin|--|
          \7 /          \7 /

$$x_{6} = 2 \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)}$$

            2/pi\        2/pi\          /pi\    /pi\
x7 = - 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--|
             \7 /         \7 /          \7 /    \7 /

$$x_{7} = - 2 \cos^{2}{\left(\frac{\pi}{7} \right)} + 2 \sin^{2}{\left(\frac{\pi}{7} \right)} - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}$$

Сумма и произведение корней

сумма

       2/pi\        2/pi\     /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\     /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\     /       /3*pi\    /pi\        /pi\    /3*pi\\        /pi\    /3*pi\        /pi\    /3*pi\     /       /pi\    /3*pi\        /3*pi\    /pi\\        /pi\    /3*pi\        /pi\    /3*pi\        /pi\          /pi\          2/pi\        2/pi\          /pi\    /pi\
- 2*cos |--| - 2*sin |--| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----| + I*|- 2*cos|----|*sin|--| + 2*cos|--|*sin|----|| - 2*cos|--|*cos|----| - 2*sin|--|*sin|----| + I*|- 2*cos|--|*sin|----| - 2*cos|----|*sin|--|| - 2*cos|--|*cos|----| + 2*sin|--|*sin|----| + 2*cos|--| + 2*I*sin|--| + - 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--|
        \7 /         \7 /     \     \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /     \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /     \       \ 7  /    \7 /        \7 /    \ 7  //        \7 /    \ 7  /        \7 /    \ 7  /     \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  /        \7 /          \7 /           \7 /         \7 /          \7 /    \7 /

$$\left(- 2 \cos^{2}{\left(\frac{\pi}{7} \right)} + 2 \sin^{2}{\left(\frac{\pi}{7} \right)} - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) + \left(\left(\left(- 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right) + \left(\left(\left(2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)\right) + \left(\left(- 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \sin^{2}{\left(\frac{\pi}{7} \right)}\right) + \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right)\right)\right) + \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right)\right)\right) + \left(2 \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)}\right)\right)$$

=

       2/pi\        /pi\     /       /pi\    /2*pi\        /2*pi\    /pi\\     /       /pi\    /3*pi\        /3*pi\    /pi\\     /       /3*pi\    /pi\        /pi\    /3*pi\\     /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /3*pi\          /pi\        /pi\    /2*pi\          /pi\    /pi\
- 4*cos |--| + 2*cos|--| + I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + I*|- 2*cos|--|*sin|----| - 2*cos|----|*sin|--|| + I*|- 2*cos|----|*sin|--| + 2*cos|--|*sin|----|| + I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 4*cos|--|*cos|----| + 2*I*sin|--| + 4*cos|--|*cos|----| - 4*I*cos|--|*sin|--|
        \7 /        \7 /     \       \7 /    \ 7  /        \ 7  /    \7 //     \       \7 /    \ 7  /        \ 7  /    \7 //     \       \ 7  /    \7 /        \7 /    \ 7  //     \     \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /          \7 /        \7 /    \ 7  /          \7 /    \7 /

$$- 4 \cos^{2}{\left(\frac{\pi}{7} \right)} - 4 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} + 4 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right) - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right) + 2 i \sin{\left(\frac{\pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right) + i \left(2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

произведение

/       2/pi\        2/pi\\ /  /     /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\\ /  /       /pi\    /2*pi\        /2*pi\    /pi\\        /pi\    /2*pi\        /pi\    /2*pi\\ /  /       /3*pi\    /pi\        /pi\    /3*pi\\        /pi\    /3*pi\        /pi\    /3*pi\\ /  /       /pi\    /3*pi\        /3*pi\    /pi\\        /pi\    /3*pi\        /pi\    /3*pi\\ /     /pi\          /pi\\ /       2/pi\        2/pi\          /pi\    /pi\\
|- 2*cos |--| - 2*sin |--||*|I*|2*cos|--|*sin|----| + 2*cos|----|*sin|--|| - 2*sin|--|*sin|----| + 2*cos|--|*cos|----||*|I*|- 2*cos|--|*sin|----| + 2*cos|----|*sin|--|| + 2*cos|--|*cos|----| + 2*sin|--|*sin|----||*|I*|- 2*cos|----|*sin|--| + 2*cos|--|*sin|----|| - 2*cos|--|*cos|----| - 2*sin|--|*sin|----||*|I*|- 2*cos|--|*sin|----| - 2*cos|----|*sin|--|| - 2*cos|--|*cos|----| + 2*sin|--|*sin|----||*|2*cos|--| + 2*I*sin|--||*|- 2*cos |--| + 2*sin |--| - 4*I*cos|--|*sin|--||
\        \7 /         \7 // \  \     \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  // \  \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  // \  \       \ 7  /    \7 /        \7 /    \ 7  //        \7 /    \ 7  /        \7 /    \ 7  // \  \       \7 /    \ 7  /        \ 7  /    \7 //        \7 /    \ 7  /        \7 /    \ 7  // \     \7 /          \7 // \        \7 /         \7 /          \7 /    \7 //

$$\left(- 2 \cos^{2}{\left(\frac{\pi}{7} \right)} - 2 \sin^{2}{\left(\frac{\pi}{7} \right)}\right) \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{2 \pi}{7} \right)} + 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{2 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{2 \pi}{7} \right)}\right)\right) \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} - 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)\right) \left(- 2 \cos{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)} + 2 \sin{\left(\frac{\pi}{7} \right)} \sin{\left(\frac{3 \pi}{7} \right)} + i \left(- 2 \sin{\left(\frac{3 \pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)} - 2 \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{3 \pi}{7} \right)}\right)\right) \left(2 \cos{\left(\frac{\pi}{7} \right)} + 2 i \sin{\left(\frac{\pi}{7} \right)}\right) \left(- 2 \cos^{2}{\left(\frac{\pi}{7} \right)} + 2 \sin^{2}{\left(\frac{\pi}{7} \right)} - 4 i \sin{\left(\frac{\pi}{7} \right)} \cos{\left(\frac{\pi}{7} \right)}\right)$$

=

-128

$$-128$$

Численный ответ

x1 = -1.24697960371747 + 1.56366296493606*i

x2 = 1.80193773580484 - 0.867767478235116*i

x3 = 0.445041867912629 - 1.94985582436365*i

x4 = -1.24697960371747 - 1.56366296493606*i

x5 = -2.0

x6 = 0.445041867912629 + 1.94985582436365*i

x7 = 1.80193773580484 + 0.867767478235116*i