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

Theorem

The roots of the polynomial:

$z^4 + 81$

are:

$\set {3 \cis 45 \degrees, 3 \cis 135 \degrees, 3 \cis 225 \degrees, 3 \cis 315 \degrees}$

Proof

From Factorisation of $z^n + 1$:

$z^4 + 1 = \ds \prod_{k \mathop = 0}^3 \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} 4}$

Thus:

$z = \set {\exp \dfrac {\paren {2 k + 1} i \pi} 4}$

\(\text {(k = 0)}: \quad\)

\(\ds z\)

\(=\)

\(\ds \cos \dfrac \pi 4 + i \sin \dfrac \pi 4\)

\(\text {(k = 1)}: \quad\)

\(\ds z\)

\(=\)

\(\ds \cos \dfrac {3 \pi} 4 + i \sin \dfrac {3 \pi} 4\)

\(\text {(k = 2)}: \quad\)

\(\ds z\)

\(=\)

\(\ds \cos \dfrac {5 \pi} 4 + i \sin \dfrac {5 \pi} 4\)

\(\text {(k = 3)}: \quad\)

\(\ds z\)

\(=\)

\(\ds \cos \dfrac {7 \pi} 4 + i \sin \dfrac {7 \pi} 4\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Roots of Complex Numbers: $97 \ \text{(a)}$