Roots of Complex Number/Examples/z^4 - 81 = 0

Theorem

The roots of the polynomial:

$z^4 - 81$

are:

$\set {3, 3 i, -3, -3 i}$

Proof

$z^4 - a = \ds \prod_{k \mathop = 0}^3 \paren {z - \alpha^k b}$

where:

$\alpha$ is a primitive complex $4$th root of unity
$b$ is any complex number such that $b^4 = a$.

Here we can take $b = 3$, as $81 = 3^4$.

Thus:

$z = \set {3 \exp \dfrac {k i \pi} 2}$

 $\text {(k = 0)}: \quad$ $\ds z$ $=$ $\ds 3 \paren {\cos \dfrac 0 \pi 2 + i \sin \dfrac 0 \pi 2}$ $\ds$ $=$ $\ds 3$ $\text {(k = 1)}: \quad$ $\ds z$ $=$ $\ds \cos \dfrac {\pi} 2 + i \sin \dfrac {\pi} 2$ $\ds$ $=$ $\ds 3 i$ $\text {(k = 2)}: \quad$ $\ds z$ $=$ $\ds \cos \dfrac {2 \pi} 2 + i \sin \dfrac {2 \pi} 2$ $\ds$ $=$ $\ds -3$ $\text {(k = 3)}: \quad$ $\ds z$ $=$ $\ds \cos \dfrac {3 \pi} 2 + i \sin \dfrac {3 \pi} 2$ $\ds$ $=$ $\ds -3 i$

$\blacksquare$