Find all solutions of the equation. Leave answers in trigonometric form.
-$$x ^ { 3 } - 8 = 0$$
A) $\left\{ 2,2 \operatorname { cis } 120 ^ { \circ } , 2 \operatorname { cis } 240 ^ { \circ } \right\}$
B) $\left\{ 2,2 \operatorname { cis } 60 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$
C) $\left\{ 2,2 \operatorname { cis } 300 ^ { \circ } , 2 \operatorname { cis } 300 ^ { \circ } \right\}$
D) $\left\{ 2,2 \right.$ cis $60 ^ { \circ } , 2$ cis $\left. 240 ^ { \circ } \right\}$

Question

Quizplus · Accepted Answer

The equation $x^3 - 8 = 0$ can be solved by adding 8 to both sides and then taking the cube root, resulting in $x = 2$. This is the real root. For the complex roots, we use the fact that the argument (angle) between each root in the complex plane is $360^\circ / n$ where $n$ is the degree of the polynomial, here $n=3$, so each root is $120^\circ$ apart. Starting from the real root at $0^\circ$, the complex roots are at $120^\circ$ and $240^\circ$, both with a magnitude of 2, hence the solutions are $2$, $2 \operatorname{cis} 120^\circ$, and $2 \operatorname{cis} 240^\circ$.

Find All Solutions of the Equation x3−8=0x ^ { 3 } - 8 = 0x3−8=0

Find All Solutions of the Equation $x ^ { 3 } - 8 = 0$