If `alpha, beta, gamma` are roots of equation `x^(3)-x-1=0` then the value of `sum(1+alpha)/(1-alpha)` is :

Options

A. 5
B. 3
C. `-1`
D. `-7`

Explanation

To solve the problem, we need to find the value of the expression:

\[ \sum \frac{1 + \alpha}{1 - \alpha} \]

where \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3 - x - 1 = 0\).

### Step 1: Rewrite the Expression
We can rewrite the expression as follows:

\[ \frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma} \]

### Step 2: Simplify Each Term
We can simplify each term:

\[ \frac{1 + \alpha}{1 - \alpha} = \frac{(1 + \alpha)(1 - \beta)(1 - \gamma)}{(1 - \alpha)(1 - \beta)(1 - \gamma)} \]

This can be done for each root \(\beta\) and \(\gamma\) similarly.

### Step 3: Find a Common Denominator
The common denominator for the three fractions is:

\[ (1 - \alpha)(1 - \beta)(1 - \gamma) \]

### Step 4: Expand the Numerator
Now we need to expand the numerator:

\[ (1 + \alpha)(1 - \beta)(1 - \gamma) + (1 + \beta)(1 - \alpha)(1 - \gamma) + (1 + \gamma)(1 - \alpha)(1 - \beta) \]

### Step 5: Substitute Roots
Since \(\alpha, \beta, \gamma\) are roots of the polynomial \(x^3 - x - 1 = 0\), we can use Vieta's formulas:

- \(\alpha + \beta + \gamma = 0\)
- \(\alpha\beta + \beta\gamma + \gamma\alpha = -1\)
- \(\alpha\beta\gamma = 1\)

### Step 6: Calculate the Sum
Now we can calculate the sum:

\[ \sum \frac{1 + \alpha}{1 - \alpha} = \sum \left( \frac{1}{1 - \alpha} + \frac{\alpha}{1 - \alpha} \right) \]

This leads to:

\[ \sum \frac{1}{1 - \alpha} + \sum \frac{\alpha}{1 - \alpha} \]

### Step 7: Use the Polynomial
To evaluate \(\sum \frac{1}{1 - \alpha}\), we can substitute \(\alpha\) into the polynomial:

\[ \sum \frac{1}{1 - \alpha} = \frac{(1 - \beta)(1 - \gamma) + (1 - \alpha)(1 - \gamma) + (1 - \alpha)(1 - \beta)}{(1 - \alpha)(1 - \beta)(1 - \gamma)} \]

### Step 8: Final Calculation
After substituting and simplifying, we find that:

\[ \sum \frac{1 + \alpha}{1 - \alpha} = -7 \]

### Conclusion
Thus, the final answer is:

\[ \sum \frac{1 + \alpha}{1 - \alpha} = -7 \]