I can't go further. Can't we find x except De-Moivre's formula?
This is much easier if you turn to a way of representing complex numbers: A complex number can be represented using real and imaginary parts, or using modulus and angle:
`x=a+bi = re^(i theta)`
Think of it as the difference between rectangular coordinates and polar coordinates. We write `x=r e^(i theta)`, where r is a nonnegative real number.
Also, `r e^(i theta) = r e^(i theta + 2 pi i)`
It's like rotating around the circle an extra time, and still landing in the same spot.
`x^7 = (r e^(i theta))^7 = r^7 e^(7 i theta)`
If we want `x^7 = 1`, then we need `r^7 = 1` and `7 theta = 2 pi` (or 0, or `4 pi`, or `6 pi`, etc).
So, we need `r=1`, and `theta = 0, (2 pi)/7, (4 pi)/7, (6 pi)/7, (8 pi)/7, (10 pi)/7, (12 pi)/7`
So, the possibilities for x are:
`x=e^(0 pi i / 7)=1`
`x=e^(2 pi i / 7)`
`x=e^(4 pi i / 7)`
`x=e^(6 pi i / 7)`
`x=e^(8 pi i / 7)`
`x=e^(10 pi i / 7)`
`x=e^(12 pi i / 7)`
`x=e^(14 pi i / 7)=e^(2 pi i )=1`
An easier was to write this is, say `delta =e^(2 pi i / 7)`
Then the possibilites for x are:
`1, delta, delta^2, delta^3, delta^4, delta^5, delta^6`
If you want to factor, as you were doing before, you would get: