If `alpha, beta, gamma` are the zeros of the cubic polynomial `ax^3+bx^2+cx+d`, then `alphabeta+betagamma+gammaalpha` is equal to:

Options

A. `-b/a`
B. `b/a`
C. `c/a`
D. `d/a`

Answer

To find the value of \( \alpha\beta + \beta\gamma + \gamma\alpha \) for the cubic polynomial \( ax^3 + bx^2 + cx + d \) where \( \alpha, \beta, \gamma \) are the zeros, we can use Vieta's formulas.

### Step-by-Step Solution:

1. **Identify the polynomial and its coefficients**:
The given polynomial is \( ax^3 + bx^2 + cx + d \). Here, the coefficients are \( a, b, c, d \).

2. **Recall Vieta's Formulas**:
According to Vieta's formulas for a cubic polynomial, we have:
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \)
- The product of the roots \( \alpha\beta\gamma = -\frac{d}{a} \)

3. **Find the required expression**:
We are specifically interested in the expression \( \alpha\beta + \beta\gamma + \gamma\alpha \).

4. **Apply Vieta's Formula**:
From Vieta's, we know:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]

5. **Conclusion**:
Therefore, the value of \( \alpha\beta + \beta\gamma + \gamma\alpha \) is \( \frac{c}{a} \).

### Final Answer:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]