PX153 - B3 - de moivre's theorem

(\cos θ + i \sin θ)^{n} = \cos n θ + i \sin n θ

follows from euler's formula and leads to

z^{n} = r^{n} e^{i n θ} = r^{n} (\cos n θ + i \sin n θ)

- valid for $n \in Z$

solving polynomials

the total number of roots = largest power
let's write the roots of an nth order polynomial as $z_{k}$ , where $k = 0, 1, 2. . . (n - 1)$ and take a polynomial of the form $z_{k}^{n} = A^{- Z}$
$A = r e^{i θ} e^{i 2 π k}$
$e^{i 2 π} = 1$ and, $e^{i 2 π k} = 1^{k} = 1$
$A^{\frac{1}{n}} = z_{k} = r^{\frac{1}{n}} e^{i \frac{θ}{n}} e^{i \frac{2 π k}{n}}$
if $k = 0$ : $z_{0} = r^{1 / n} e^{i θ / n}$
if $k = 1$ : $z_{1} = r^{1 / n} e^{i θ / n} e^{i 2 π / n} = z_{0} e^{i 2 π / n}$
...
if $k = n - 1$ : $z_{n - 1} = r^{1 / n} e^{i θ / n} e^{i (n - 1) 2 π / n}$
$$z_n=z_0e^{ik2\pi/n}=z_0e^{i2\pi}=z_0$$
n roots from $k = 0$ to $k = n - 1$
eg: solve $z^{4} - 1 = - i \sqrt{3}$
eg: solve $z^{3} - (2 - 2 i) z = 9$
eg: solve $z^{4} - (2 - 2 i) z^{3} - (2 + 2 i) z + 8 = 0$