Question

If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then the value of \( \alpha^3 + \beta^3 \) is:

• A. \( 3ab + b^3 \)
• B. \( \frac{a^3 + b^3}{a^3} \)
• C. \( 3ab + b^3 \)
• D. \( \frac{3ab + b^3}{a^3} \)

Hint:

For cubic equations, use the identity \( \alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha\beta] \) to simplify the calculation.

Answer 1

The Correct Option is C

Step 1: Use the identity. We use the identity \( \alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha\beta] \). The sum and product of the roots can be calculated using Vieta’s formulas, where \( \alpha + \beta = -\frac{b}{a} \) and \( \alpha\beta = \frac{c}{a} \).
Step 2: Conclusion. Thus, the value of \( \alpha^3 + \beta^3 \) is \( 3ab + b^3 \).