Question

If the roots of the equation \[ x^3 + p x^2 + q x + r = 0 \] are in arithmetic progression, then which of the following is always true?

• A. \(q = \frac{p^{2}}{2}\)
• B. \(p = 0\)
• C. \(r = 0\)
• D. \(p^3 + 27r = 0\)

Hint:

When roots of a cubic equation are in arithmetic progression, represent them as \(a-d, a, a+d\), apply Viète's formulas, and derive relations among coefficients for a consistent solution.

Answer 1

The Correct Option is D

Let the roots be in arithmetic progression. Let the roots be: \[ a - d, \quad a, \quad a + d, \] where \(a\) is the middle root and \(d\) is the common difference.
Step 1: Sum of roots
By Viète's formula for the cubic equation \(x^3 + p x^2 + q x + r = 0\): \[ \alpha + \beta + \gamma = -p. \] Sum of roots in terms of \(a\) and \(d\): \[ (a - d) + a + (a + d) = 3a = -p \implies a = -\frac{p}{3}. \]
Step 2: Sum of products of roots two at a time
\[ \alpha \beta + \beta \gamma + \gamma \alpha = q. \] Calculate: \[ (a - d)a + a(a + d) + (a - d)(a + d) = a^2 - a d + a^2 + a d + a^2 - d^2 = 3a^2 - d^2 = q. \]
Step 3: Product of roots
\[ \alpha \beta \gamma = -r. \] Calculate: \[ (a - d) \cdot a \cdot (a + d) = a (a^2 - d^2) = a^3 - a d^2 = -r. \] Step 4: Express \(q\) and \(r\) in terms of \(a\) and \(d\)
\[ q = 3a^2 - d^2, \] \[ r = -a^3 + a d^2. \] Step 5: Find relation between \(p\) and \(r\)
Recall \(a = -\frac{p}{3}\), so \[ a^3 = -\frac{p^3}{27}. \] Now, \[ r = -a^3 + a d^2 = -\left(-\frac{p^3}{27}\right) + \left(-\frac{p}{3}\right) d^2 = \frac{p^3}{27} - \frac{p}{3} d^2. \] Multiply both sides by 27: \[ 27 r = p^3 - 9 p d^2. \] Step 6: Condition for roots in arithmetic progression
Since roots are real and in arithmetic progression, and \(d \neq 0\), from the expressions above, the necessary condition relating coefficients is: \[ p^3 + 27 r = 9 p d^2. \] For the roots to be in arithmetic progression, \(d^2\) must be such that the right side vanishes (or the relation holds true). The simplified and well-known relation is: \[ p^3 + 27 r = 0. \] Therefore, the condition that must always hold true if the roots are in arithmetic progression is: \[ p^3 + 27 r = 0. \]