Question

If \( x^2 + x + 1 = 0 \), then \[ \left( x + \frac{1}{x} \right)^4 + \left( x^2 + \frac{1}{x^2} \right)^4 + \left( x^3 + \frac{1}{x^3} \right)^4 + \dots + \left( x^{25} + \frac{1}{x^{25}} \right)^4 \] is

Hint:

For problems involving cube roots of unity, recognize the repeating patterns and simplify the powers of \( x \) accordingly.

Answer 1

Correct Answer: 145

Step 1: Use the given equation.
We are given \( x^2 + x + 1 = 0 \). The solutions to this equation are the cube roots of unity. Let \( \omega \) be a cube root of unity. This means that: \[ \omega^2 + \omega + 1 = 0 \] and the cube roots of unity satisfy \( \omega^3 = 1 \). Step 2: Express powers in terms of \( \omega \).
We now express powers of \( x \) in terms of \( \omega \), as the expression \( x + \frac{1}{x} \) will have a cyclic property for the cube roots of unity. Step 3: Compute the sum of terms.
The terms of the form \( \left( x^k + \frac{1}{x^k} \right)^4 \) for \( k = 1, 2, 3, \dots, 25 \) will simplify using properties of the cube roots of unity. By calculating the sum of these terms, we find the value to be 145.