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: \[ 9cb = 2b^3 + 27d. \]

Answer 2

Problem: Find the condition that the roots of the cubic equation \[ x^3 - b x^2 + c x - d = 0 \] are in arithmetic progression.

Solution: Let the three roots be \( \alpha, \beta, \gamma \) in arithmetic progression. Then, \[ \beta = \alpha + r, \quad \gamma = \alpha + 2r, \] for some common difference \( r \).

Using Viète's formulas: \[ \alpha + \beta + \gamma = b, \] \[ \alpha \beta + \beta \gamma + \gamma \alpha = c, \] \[ \alpha \beta \gamma = d. \] Substitute the roots: \[ \alpha + (\alpha + r) + (\alpha + 2r) = 3\alpha + 3r = b \implies \alpha + r = \frac{b}{3}. \] Express \( \alpha = \frac{b}{3} - r \). Sum of products of roots two at a time: \[ \alpha \beta + \beta \gamma + \gamma \alpha = c. \] Calculate each term: \[ \alpha \beta = \alpha (\alpha + r) = \alpha^2 + \alpha r, \] \[ \beta \gamma = (\alpha + r)(\alpha + 2r) = \alpha^2 + 3\alpha r + 2r^2, \] \[ \gamma \alpha = (\alpha + 2r) \alpha = \alpha^2 + 2\alpha r. \] Sum: \[ 3 \alpha^2 + 6 \alpha r + 2 r^2 = c. \] Similarly, product of roots: \[ \alpha \beta \gamma = \alpha (\alpha + r)(\alpha + 2r) = d. \] Expand: \[ \alpha (\alpha^2 + 3\alpha r + 2 r^2) = \alpha^3 + 3 \alpha^2 r + 2 \alpha r^2 = d. \] Using the relation \( \alpha + r = \frac{b}{3} \), substitute \( \alpha = \frac{b}{3} - r \) and simplify the above equations. After algebraic manipulation, the condition for roots in arithmetic progression reduces to: \[ 9 c b = 2 b^3 + 27 d. \]

Final answer: \[ \boxed{9 c b = 2 b^3 + 27 d}. \]