If `alpha, beta in C` are the distinct roots of the equation `x^(2)-x+1=0`, then `alpha^(101)+beta^(107)` is equal to

Options

A. 2
B. `-1`
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \( \alpha^{101} + \beta^{107} \), where \( \alpha \) and \( \beta \) are the distinct roots of the equation \( x^2 - x + 1 = 0 \).

### Step 1: Find the roots of the equation
The roots of the quadratic equation \( x^2 - x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = 1 \). Plugging in these values:
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]
Thus, the roots are:
\[ \alpha = \frac{1 + i\sqrt{3}}{2}, \quad \beta = \frac{1 - i\sqrt{3}}{2} \]

### Step 2: Express powers of roots in terms of \( \omega \)
We can express \( \alpha \) and \( \beta \) in terms of the cube roots of unity. Let \( \omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} \) and \( \omega^2 = e^{-2\pi i / 3} = \frac{-1 - i\sqrt{3}}{2} \). Notice that:
\[ \alpha = -\omega, \quad \beta = -\omega^2 \]

### Step 3: Calculate \( \alpha^{101} + \beta^{107} \)
Now we calculate:
\[ \alpha^{101} + \beta^{107} = (-\omega)^{101} + (-\omega^2)^{107} \]
This simplifies to:
\[ (-1)^{101} \cdot \omega^{101} + (-1)^{107} \cdot \omega^{214} \]
Since \( (-1)^{101} = -1 \) and \( (-1)^{107} = -1 \), we have:
\[ -\omega^{101} - \omega^{214} \]

### Step 4: Reduce the powers using \( \omega^3 = 1 \)
We can reduce the powers of \( \omega \):
\[ \omega^{101} = \omega^{3 \cdot 33 + 2} = \omega^2 \]
\[ \omega^{214} = \omega^{3 \cdot 71 + 1} = \omega \]
Thus, we have:
\[ -\omega^{101} - \omega^{214} = -\omega^2 - \omega \]

### Step 5: Use the property of roots of unity
We know from the properties of cube roots of unity that:
\[ 1 + \omega + \omega^2 = 0 \implies \omega + \omega^2 = -1 \]
Therefore:
\[ -\omega^2 - \omega = -(-1) = 1 \]

### Final Answer
Thus, the value of \( \alpha^{101} + \beta^{107} \) is:
\[ \boxed{1} \]