Question

The condition that the roots of \( x^3 - bx^2 + cx - d = 0 \) are in arithmetic progression is:

• A. \( 9cb = 2b^3 + 27d \)
• B. \( 9cb = 2d^3 + 27b \)
• C. \( 9cd = 2b^3 + 27d \)
• D. \( 9cd = 2d^3 + 27b \)

Hint:

For problems involving roots in arithmetic progression, use Vieta’s relations to express sums and products of the roots and solve for the conditions involving the coefficients.

Answer 1

The Correct Option is A

Step 1: Vieta's Relations. For the cubic equation \( x^3 - bx^2 + cx - d = 0 \), the relations from Vieta's formulas give: \[ r_1 + r_2 + r_3 = b, \quad r_1r_2 + r_2r_3 + r_3r_1 = c, \quad r_1r_2r_3 = d. \] Step 2: Roots in Arithmetic Progression. Let the roots be in arithmetic progression. Thus, we assume \( r_1 = r_2 - d \), \( r_2 = r_2 \), and \( r_3 = r_2 + d \). From the relations, we get the condition: \[ 9cb = 2b^3 + 27d. \]