Question

Consider the equation \( x^3 - 1 = 0 \). If one of the solutions to this equation is 1, the other solution(s) is/are

• A. \( -\frac{1}{2} + \frac{\sqrt{3}}{2} i \)
• B. \( i \)
• C. \( -i \)
• D. \( -\frac{1}{2} - \frac{\sqrt{3}}{2} i \)

Hint:

The cube roots of unity are 1, \( \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \), and \( \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \). These roots satisfy the equation \( x^3 = 1 \).

Answer 1

The Correct Option is A, D

Step 1: Understanding the roots of the equation. 
The equation \( x^3 - 1 = 0 \) can be factored as \( (x - 1)(x^2 + x + 1) = 0 \). The roots of this equation are \( x = 1 \) and the two roots of the quadratic equation \( x^2 + x + 1 = 0 \). These roots are the cube roots of unity, which are: \[ x = \omega = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \text{and} x = \omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2} i \] where \( \omega \) is a primitive cube root of unity.

Step 2: Analyzing the options. 
(A) This is \( \omega \), which is one of the solutions. 
(B) \( i \) is not a solution to the equation. 
(C) \( -i \) is also not a solution. 
(D) This is \( \omega^2 \), the other solution to the equation. 
 

Step 3: Conclusion. 
The correct answer is

(A) \( -\frac{1}{2} + \frac{\sqrt{3}}{2} i \) 

(D) \( -\frac{1}{2} - \frac{\sqrt{3}}{2} i \)