If `alpha, beta, gamma` are the roots of the equation `x^3 + px^2 + qx + r = n` then the value of `(alpha - 1/(beta gamma)) (beta -1/(gamma alpha)) (gamma-1/(alpha beta))` is:

Answer

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

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

where \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3 + px^2 + qx + r = n\).

### Step 1: Rewrite the equation
We start with the equation:

\[ x^3 + px^2 + qx + r - n = 0 \]

This implies that the roots \(\alpha, \beta, \gamma\) satisfy the polynomial equation \(x^3 + px^2 + qx + (r - n) = 0\).

### Step 2: Use Vieta's Formulas
From Vieta's formulas, we know:

- \(\alpha + \beta + \gamma = -p\)
- \(\alpha \beta + \beta \gamma + \gamma \alpha = q\)
- \(\alpha \beta \gamma = -(r - n)\)

### Step 3: Simplify the expression
Now we simplify the expression:

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

Substituting \(\beta \gamma = \frac{\alpha \beta \gamma}{\alpha}\), \(\gamma \alpha = \frac{\alpha \beta \gamma}{\beta}\), and \(\alpha \beta = \frac{\alpha \beta \gamma}{\gamma}\):

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

This simplifies to:

\[ = \left(\alpha - \frac{\alpha}{\alpha \beta \gamma}\right)\left(\beta - \frac{\beta}{\alpha \beta \gamma}\right)\left(\gamma - \frac{\gamma}{\alpha \beta \gamma}\right) \]

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

### Step 4: Factor out common terms
Now, factor out \(\alpha \beta \gamma\):

\[ = \frac{(\alpha \beta \gamma)^3 - (\alpha + \beta + \gamma)}{\alpha \beta \gamma} \]

### Step 5: Substitute the values
Substituting the values from Vieta's formulas:

\[ = \frac{(-(r - n))^3 - (-p)}{-(r - n)} \]

This simplifies to:

\[ = \frac{(r - n)^3 + p}{r - n} \]

### Final Result
Thus, the value of the expression is:

\[ = (r - n)^2 + \frac{p}{r - n} \]