If `f(x)=a=b x+c x^2a n dalpha,beta,gamma` are the roots of the equation `x^3=1,t h e n|a b c b c a c a b|` is equal to `f(alpha)+f(beta)+f(gamma)` `f(alpha)f(beta)+f(beta)f(gamma)+f(gamma)f(alpha)` `f(alpha)f(beta)f(gamma)` `-f(alpha)f(beta)f(gamma)`

Options

A. `f(alpha) +f(beta) +f(gamma)`
B. `f(alpha) f(beta)+f(beta)f(gamma)+f(gamma) f(alpha)`
C. `f(alpha) f(beta) f(gamma)`
D. `-f(alpha) f(beta)f(gamma)`

Explanation

To solve the given problem, we need to analyze the function \( f(x) = ax + bx + cx^2 \) and the roots of the equation \( x^3 = 1 \), which are \( \alpha, \beta, \gamma \). The roots of this equation are the cube roots of unity: \( \omega, \omega^2, 1 \), where \( \omega = e^{2\pi i / 3} \).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the equation \( x^3 = 1 \) are:
\[ \alpha = 1, \quad \beta = \omega, \quad \gamma = \omega^2 \]

2. **Evaluate \( f(\alpha), f(\beta), f(\gamma) \)**:
We need to evaluate \( f \) at each of the roots:
- For \( \alpha = 1 \):
\[ f(1) = a(1) + b(1) + c(1^2) = a + b + c \]
- For \( \beta = \omega \):
\[ f(\omega) = a(\omega) + b(\omega) + c(\omega^2) = a\omega + b\omega + c\omega^2 \]
- For \( \gamma = \omega^2 \):
\[ f(\omega^2) = a(\omega^2) + b(\omega^2) + c(\omega) = a\omega^2 + b\omega^2 + c\omega \]

3. **Use the Properties of Roots of Unity**:
The sum of the roots \( \alpha + \beta + \gamma = 1 + \omega + \omega^2 = 0 \). This property will help simplify our calculations.

4. **Calculate the Determinant**:
We need to calculate the determinant:
\[ |a \quad b \quad c| \]
\[ |b \quad c \quad a| \]
\[ |c \quad a \quad b| \]
Using the determinant properties, we can simplify this to:
\[ = abc - (a^3 + b^3 + c^3 - 3abc) \]
This can be factored as:
\[ = - (a + b + c)(a + b\omega + c\omega^2)(a + b\omega^2 + c\omega) \]

5. **Final Result**:
The determinant can be expressed in terms of \( f(\alpha), f(\beta), f(\gamma) \):
\[ = -f(\alpha)f(\beta)f(\gamma) \]
Therefore, the correct option is:
\[ -f(\alpha)f(\beta)f(\gamma) \]

### Conclusion:
The answer to the problem is:
\[ \text{The value of the determinant is } -f(\alpha)f(\beta)f(\gamma). \]