Question

If \( \alpha \) and \( \beta \) are the roots of \( x^2 - x + 1 = 0 \), then the equation whose roots are \( \alpha^{100} \) and \( \beta^{100} \) is

• A. \( x^2 - x + 1 = 0 \)
• B. \( x^2 + x + 1 = 0 \)
• C. \( x^2 - x - 1 = 0 \)
• D. \( x^2 + x - 1 = 0 \)

Hint:

The roots of unity cycle periodically. In this case, the powers of the roots repeat with a period of 3.

Answer 1

The Correct Option is B

Step 1: Use properties of roots of unity.
Since the roots of \( x^2 - x + 1 = 0 \) are cube roots of unity, we know that \( \alpha^{100} \) and \( \beta^{100} \) will satisfy the same equation as \( \alpha \) and \( \beta \).
Step 2: Conclusion.
The equation whose roots are \( \alpha^{100} \) and \( \beta^{100} \) is \( x^2 + x + 1 = 0 \). Final Answer: \[ \boxed{x^2 + x + 1 = 0} \]