If `alpha, beta, gamma` are the roots of the equation `x^(3) + ax^(2) + bx + c = 0, "then" alpha^(-1) + beta^(-1) + gamma^(-1)=`

Options

A. `(a)/(c)`
B. `-(b)/(c)`
C. `(b)/(a)`
D. `(c)/(a)`

Answer

To find the value of \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \) given that \( \alpha, \beta, \gamma \) are the roots of the cubic equation \( x^3 + ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Understand the relationship between roots and coefficients
For a cubic polynomial of the form \( x^3 + ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -a \)
- The sum of the product of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = b \)
- The product of the roots \( \alpha\beta\gamma = -c \)

### Step 2: Rewrite the expression for the reciprocals of the roots
We need to find:
\[ \alpha^{-1} + \beta^{-1} + \gamma^{-1} \]
This can be rewritten using a common denominator:
\[ \alpha^{-1} + \beta^{-1} + \gamma^{-1} = \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\beta\gamma + \gamma\alpha + \alpha\beta}{\alpha\beta\gamma} \]

### Step 3: Substitute the known values
From the relationships established in Step 1:
- The numerator \( \beta\gamma + \gamma\alpha + \alpha\beta \) is equal to \( b \).
- The denominator \( \alpha\beta\gamma \) is equal to \( -c \).

Thus, we can substitute these values into the expression:
\[ \alpha^{-1} + \beta^{-1} + \gamma^{-1} = \frac{b}{-c} = -\frac{b}{c} \]

### Final Answer
Therefore, the value of \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \) is:
\[ -\frac{b}{c} \]

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