Question

If \( \alpha, \beta, \gamma \) are roots of the equation \( x^3 + ax^2 + bx + c = 0 \), then \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \) is:

• A. \( \frac{a}{c} \)
• B. \( \frac{-b}{c} \)
• C. \( \frac{c}{a} \)
• D. \( \frac{b}{a} \)

Hint:

For cubic equations with roots, use Vieta’s formulas to relate the coefficients of the equation to sums and products of the roots.

Answer 1

The Correct Option is B

We are given the cubic equation \( x^3 + ax^2 + bx + c = 0 \) with roots \( \alpha, \beta, \gamma \). By Vieta's formulas, we know: - \( \alpha + \beta + \gamma = -a \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = b \)
- \( \alpha\beta\gamma = -c \)

 Step 1: We need to find the value of \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} \). Using the identity: \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} = \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma} \) 

 

Step 2: Substitute the values from Vieta’s formulas: \( \alpha^{-1} + \beta^{-1} + \gamma^{-1} = \frac{b}{-c} = \frac{-b}{c} \)