Question

If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 2x + 5 = 0 \), then \( \sum \frac{\beta + \gamma}{\alpha^2} \) is:

• A. \( \frac{-2}{5} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{2}{5} \)
• D. \( \frac{-3}{5} \)

Hint:

For expressions involving sums of roots, use Vieta's relations to express everything in terms of the coefficients of the polynomial.

Answer 1

The Correct Option is C

We are given that \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + 2x + 5 = 0 \). According to Vieta's formulas: \[ \alpha + \beta + \gamma = 0, \quad \alpha\beta + \beta\gamma + \gamma\alpha = 2, \quad \alpha\beta\gamma = -5 \] We need to find \( \sum \frac{\beta + \gamma}{\alpha^2} \). Using the fact that \( \beta + \gamma = -\alpha \), we have: \[ \sum \frac{\beta + \gamma}{\alpha^2} = \frac{-\alpha}{\alpha^2} = \frac{-1}{\alpha} \] Thus, the value of \( \sum \frac{\beta + \gamma}{\alpha^2} = \frac{2}{5} \).