Question

If \( a_1, a_2, \dots, a_8 \) are the roots of the equation \( x^8 + x^7 + \dots + x + 1 = 0 \), then the value of \( a_1^{2025} + a_2^{2025} + \dots + a_8^{2025} \) is

• A. 0
• B. 8
• C. 4
• D. 2

Hint:

Recognizing that the polynomial \(1+x+\dots+x^{n-1}\) is related to the roots of unity is a crucial shortcut. The roots of this polynomial are the \((n+1)\)-th roots of unity, except for 1.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The given polynomial is a geometric series. We can use the formula for the sum of a geometric series to find a simpler equation that the roots must satisfy. This will reveal the nature of the roots (roots of unity).
Step 2: Key Formula or Approach:
1. The sum of a geometric series \( 1 + x + x^2 + \dots + x^{n-1} \) is \( \frac{x^n - 1}{x - 1} \). 2. If \( \omega \) is an n-th root of unity, then \( \omega^n = 1 \). 3. We need to evaluate \( \sum_{k=1}^{8} a_k^{2025} \). We can simplify \( a_k^{2025} \) using the properties of the roots.
Step 3: Detailed Explanation:
The given equation is \( x^8 + x^7 + \dots + x + 1 = 0 \). This is the sum of a geometric progression. For \( x \neq 1 \), we can write the sum as: \[ \frac{x^9 - 1}{x - 1} = 0 \] This implies that \( x^9 - 1 = 0 \), with the condition that \( x \neq 1 \). So, the roots \( a_1, a_2, \dots, a_8 \) are the 9th roots of unity, excluding the root 1. This means that for each root \( a_k \), we have the property: \[ a_k^9 = 1 \] We need to find the value of the sum \( S = a_1^{2025} + a_2^{2025} + \dots + a_8^{2025} \). Let's analyze the exponent, 2025. We can check its relationship with 9. A number is divisible by 9 if the sum of its digits is divisible by 9. Sum of digits of 2025 = \( 2 + 0 + 2 + 5 = 9 \). Since the sum of digits is 9, 2025 is divisible by 9. Let's find the quotient: \( 2025 \div 9 = 225 \). So, \( 2025 = 9 \times 225 \). Now we can simplify each term in the sum: \[ a_k^{2025} = a_k^{9 \times 225} = (a_k^9)^{225} \] Since \( a_k^9 = 1 \), we have: \[ a_k^{2025} = (1)^{225} = 1 \] This is true for every root \( a_k \) from \( k=1 \) to 8. Therefore, the sum becomes: \[ S = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 \] There are 8 terms in the sum. \[ S = 8 \] Step 4: Final Answer:
The value of the sum is 8.