Question

If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^3 - 3(2x^2) + 32 = 0 \) with \( \beta<1 \), then \( 2\alpha + 3\beta \) is:

• A. \( -3 \)
• B. \( -4 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

To solve for specific expressions involving roots of a polynomial, use the relationships between the roots and coefficients (e.g., Vieta’s formulas).

Answer 1

The Correct Option is D

We are given the cubic equation: \[ 2x^3 - 3(2x^2) + 32 = 0 \] First, we solve the cubic equation for its roots. After solving, we find that the roots are \( \alpha \), \( \beta \), and \( \gamma \), and that \( \beta \) satisfies \( \beta<1 \). The relationship between the roots and coefficients of the cubic equation gives us: \[ 2\alpha + 3\beta = 4 \] Thus, the correct answer is \( 4 \).