Question

If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + ax^2 + b = 0 \), then the value of \[ \frac{\alpha \beta}{\gamma}, \quad \frac{\beta \gamma}{\alpha}, \quad \frac{\gamma \alpha}{\beta} \]

• A. \( \frac{-a^3}{c^3} \)
• B. \( -a^3 \)
• C. \( \frac{a^3}{b^3} \)
• D. \( \frac{a^2}{b^3} \)

Hint:

Vieta's relations help relate the coefficients of a polynomial to the sums and products of its roots.

Answer 1

The Correct Option is C

Step 1: Use Vieta's relations.
Using Vieta’s formulas, we find that the given expression for the product of roots results in \( \frac{a^3}{b^3} \).
Step 2: Conclusion.
Thus, the value is \( \frac{a^3}{b^3} \).
Final Answer: \[ \boxed{\frac{a^3}{b^3}} \]