Consider `alpha, beta ,gamma` are the roots of `x^(3)-x^(2)-3x+4=0` such that `tan^(-1)alpha+tan^(-1)beta+tan^(-1)gamma=theta` . If the positive value of `tan (theta)` is p/q, where p and q are natural numbers, then find the value of `(p + q)`.

Answer

To solve the problem, we need to find the roots of the polynomial equation \( x^3 - x^2 - 3x + 4 = 0 \) and then use the properties of inverse tangent to find \( \tan(\theta) \) where \( \theta = \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma) \).

### Step 1: Find the roots of the polynomial
The given polynomial is:
\[ x^3 - x^2 - 3x + 4 = 0 \]

We can use the Rational Root Theorem to test for possible rational roots. Testing \( x = 1 \):
\[ 1^3 - 1^2 - 3(1) + 4 = 1 - 1 - 3 + 4 = 1 \quad (\text{not a root}) \]
Testing \( x = -1 \):
\[ (-1)^3 - (-1)^2 - 3(-1) + 4 = -1 - 1 + 3 + 4 = 5 \quad (\text{not a root}) \]
Testing \( x = 2 \):
\[ 2^3 - 2^2 - 3(2) + 4 = 8 - 4 - 6 + 4 = 2 \quad (\text{not a root}) \]
Testing \( x = -2 \):
\[ (-2)^3 - (-2)^2 - 3(-2) + 4 = -8 - 4 + 6 + 4 = -2 \quad (\text{not a root}) \]
Testing \( x = 4 \):
\[ 4^3 - 4^2 - 3(4) + 4 = 64 - 16 - 12 + 4 = 40 \quad (\text{not a root}) \]
Testing \( x = -4 \):
\[ (-4)^3 - (-4)^2 - 3(-4) + 4 = -64 - 16 + 12 + 4 = -64 \quad (\text{not a root}) \]
After testing several values, we can use synthetic division or numerical methods to find the roots. For simplicity, let's assume we found the roots to be \( \alpha, \beta, \gamma \).

### Step 2: Use the properties of inverse tangent
We know that:
\[ \tan^{-1}(\alpha) + \tan^{-1}(\beta) + \tan^{-1}(\gamma) = \theta \]
Using the formula for the sum of inverse tangents:
\[ \tan^{-1} x + \tan^{-1} y = \tan^{-1} \left( \frac{x+y}{1-xy} \right) \quad \text{if } xy < 1 \]
For three variables, we can express:
\[ \tan(\theta) = \frac{\alpha + \beta + \gamma - \alpha \beta \gamma}{1 - (\alpha \beta + \beta \gamma + \gamma \alpha)} \]

### Step 3: Calculate the sums and products of the roots
From Vieta's formulas:
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{-1}{1} = 1 \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = -\frac{-3}{1} = -3 \)
- The product of the roots \( \alpha \beta \gamma = -\frac{4}{1} = -4 \)

### Step 4: Substitute into the tangent formula
Now substituting these values into the tangent formula:
\[ \tan(\theta) = \frac{1 - (-4)}{1 - (-3)} = \frac{1 + 4}{1 + 3} = \frac{5}{4} \]

### Step 5: Express \( \tan(\theta) \) in the form \( \frac{p}{q} \)
Here, we have:
\[ \tan(\theta) = \frac{5}{4} \]
Thus, \( p = 5 \) and \( q = 4 \).

### Step 6: Find \( p + q \)
Finally, we calculate:
\[ p + q = 5 + 4 = 9 \]

### Final Answer
The value of \( p + q \) is \( \boxed{9} \).