If ` alpah , beta , gamma` are the roots of the equation `x^3 + qx +r=0` find the value of
`( beta + gamma )^(-1) +( gamma + alpha )^(-1) + ( alpha + beta)^(-1)`

Answer

To solve the problem, we need to find the value of \( ( \beta + \gamma )^{-1} + ( \gamma + \alpha )^{-1} + ( \alpha + \beta )^{-1} \) given that \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + qx + r = 0 \).

### Step-by-Step Solution:

1. **Identify the relationships from the roots:**
From Vieta's formulas, we know:
- The sum of the roots:
\[ \alpha + \beta + \gamma = 0 \]
- The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = q \]
- The product of the roots:
\[ \alpha\beta\gamma = -r \]

2. **Express the required sum in terms of the roots:**
We rewrite the expression:
\[ ( \beta + \gamma )^{-1} + ( \gamma + \alpha )^{-1} + ( \alpha + \beta )^{-1} \]
Using the relationships from Vieta's formulas:
- \( \beta + \gamma = -\alpha \)
- \( \gamma + \alpha = -\beta \)
- \( \alpha + \beta = -\gamma \)

Therefore, we can rewrite the expression as:
\[ (-\alpha)^{-1} + (-\beta)^{-1} + (-\gamma)^{-1} \]
This simplifies to:
\[ -\left( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} \right) \]

3. **Combine the fractions:**
The expression \( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} \) can be combined using a common denominator:
\[ \frac{\beta\gamma + \gamma\alpha + \alpha\beta}{\alpha\beta\gamma} \]
Substituting the known values from Vieta's formulas:
\[ = \frac{q}{-r} \]

4. **Final expression:**
Thus, we have:
\[ -\left( \frac{q}{-r} \right) = \frac{q}{r} \]

### Final Answer:
\[ ( \beta + \gamma )^{-1} + ( \gamma + \alpha )^{-1} + ( \alpha + \beta )^{-1} = \frac{q}{r} \]