If `x^(2) - x+ 1 =0, sum_(n=1)^(5) (x^(n) +(1)/(x^(n)))^2` equals -

Options

A. 8
B. 10
C. 12
D. 14

Explanation

To solve the problem, we need to find the value of the expression \( \sum_{n=1}^{5} \left( x^n + \frac{1}{x^n} \right)^2 \) given that \( x^2 - x + 1 = 0 \).

### Step 1: Find the roots of the quadratic equation
The roots of the equation \( x^2 - x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -1, c = 1 \). Thus,
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm i\sqrt{3}}{2} \]
Let \( \omega_1 = \frac{1 + i\sqrt{3}}{2} \) and \( \omega_2 = \frac{1 - i\sqrt{3}}{2} \).

### Step 2: Calculate \( x + \frac{1}{x} \)
To simplify the expression, we first compute \( x + \frac{1}{x} \):
\[ \frac{1}{x} = \frac{2}{1 + i\sqrt{3}} \cdot \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}} = \frac{2(1 - i\sqrt{3})}{1 + 3} = \frac{1 - i\sqrt{3}}{2} \]
Thus,
\[ x + \frac{1}{x} = \frac{1 + i\sqrt{3}}{2} + \frac{1 - i\sqrt{3}}{2} = \frac{2}{2} = 1 \]

### Step 3: Calculate \( x^2 + \frac{1}{x^2} \)
Using the identity \( x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 - 2 \):
\[ x^2 + \frac{1}{x^2} = 1^2 - 2 = -1 \]

### Step 4: Calculate \( x^3 + \frac{1}{x^3} \)
Using the identity \( x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 + \frac{1}{x^2}) - (x + \frac{1}{x}) \):
\[ x^3 + \frac{1}{x^3} = 1 \cdot (-1) - 1 = -1 - 1 = -2 \]

### Step 5: Calculate \( x^4 + \frac{1}{x^4} \)
Using the identity \( x^4 + \frac{1}{x^4} = (x^2 + \frac{1}{x^2})^2 - 2 \):
\[ x^4 + \frac{1}{x^4} = (-1)^2 - 2 = 1 - 2 = -1 \]

### Step 6: Calculate \( x^5 + \frac{1}{x^5} \)
Using the identity \( x^5 + \frac{1}{x^5} = (x + \frac{1}{x})(x^4 + \frac{1}{x^4}) - (x^3 + \frac{1}{x^3}) \):
\[ x^5 + \frac{1}{x^5} = 1 \cdot (-1) - (-2) = -1 + 2 = 1 \]

### Step 7: Calculate the sum
Now we can compute the required sum:
\[ \sum_{n=1}^{5} \left( x^n + \frac{1}{x^n} \right)^2 = \left( x + \frac{1}{x} \right)^2 + \left( x^2 + \frac{1}{x^2} \right)^2 + \left( x^3 + \frac{1}{x^3} \right)^2 + \left( x^4 + \frac{1}{x^4} \right)^2 + \left( x^5 + \frac{1}{x^5} \right)^2 \]
Calculating each term:
- \( (x + \frac{1}{x})^2 = 1^2 = 1 \)
- \( (x^2 + \frac{1}{x^2})^2 = (-1)^2 = 1 \)
- \( (x^3 + \frac{1}{x^3})^2 = (-2)^2 = 4 \)
- \( (x^4 + \frac{1}{x^4})^2 = (-1)^2 = 1 \)
- \( (x^5 + \frac{1}{x^5})^2 = 1^2 = 1 \)

Adding these values together:
\[ 1 + 1 + 4 + 1 + 1 = 8 \]

### Final Answer
The value of \( \sum_{n=1}^{5} \left( x^n + \frac{1}{x^n} \right)^2 \) is \( \boxed{8} \).