If the roots of the equation `x^(3) + bx^(2) + cx + d = 0` are in arithmetic progression, then b, c and d satisfy the relation

Options

A. `2b^2 - 27d = 9bc`
B. `2b^3 - 27d = 9bc`
C. `2b^2 + 27d = 9bc`
D. `2b^3 + 27d = 9bc`

Answer

To solve the problem, we need to establish the relationship between the coefficients \( b \), \( c \), and \( d \) of the polynomial \( x^3 + bx^2 + cx + d = 0 \) given that the roots are in arithmetic progression.

### Step-by-Step Solution

1. **Define the Roots**:
Let the roots of the polynomial be \( \alpha, \beta, \gamma \). Since the roots are in arithmetic progression, we can express them as:
\[ \alpha = a - d, \quad \beta = a, \quad \gamma = a + d \]
where \( a \) is the middle term and \( d \) is the common difference.

2. **Use Vieta's Formulas**:
According to Vieta's formulas for a cubic polynomial \( x^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -b \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \alpha\gamma = c \)
- The product of the roots \( \alpha\beta\gamma = -d \)

3. **Calculate the Sum of the Roots**:
\[ \alpha + \beta + \gamma = (a - d) + a + (a + d) = 3a \]
Therefore, from Vieta's:
\[ 3a = -b \quad \Rightarrow \quad a = -\frac{b}{3} \]

4. **Calculate the Sum of the Products of the Roots**:
\[ \alpha\beta + \beta\gamma + \alpha\gamma = (a - d)a + a(a + d) + (a - d)(a + d) \]
Simplifying each term:
- \( (a - d)a = a^2 - ad \)
- \( a(a + d) = a^2 + ad \)
- \( (a - d)(a + d) = a^2 - d^2 \)

Combining these:
\[ \alpha\beta + \beta\gamma + \alpha\gamma = (a^2 - ad) + (a^2 + ad) + (a^2 - d^2) = 3a^2 - d^2 \]
Setting this equal to \( c \):
\[ 3a^2 - d^2 = c \]

5. **Calculate the Product of the Roots**:
\[ \alpha\beta\gamma = (a - d)a(a + d) = a(a^2 - d^2) = -d \]
Thus:
\[ a^3 - ad^2 = -d \]

6. **Substituting \( a = -\frac{b}{3} \)**:
Substitute \( a \) into the equations:
- For \( c \):
\[ 3\left(-\frac{b}{3}\right)^2 - d^2 = c \quad \Rightarrow \quad \frac{b^2}{3} - d^2 = c \]
Rearranging gives:
\[ d^2 = \frac{b^2}{3} - c \]

- For \( d \):
\[ \left(-\frac{b}{3}\right)^3 - \left(-\frac{b}{3}\right)d^2 = -d \]
Rearranging gives:
\[ \frac{b^3}{27} + \frac{bd^2}{3} = -d \]

7. **Final Relation**:
After manipulating the equations, we arrive at the relationship:
\[ 2b^3 + 27d = 9bc \]

### Conclusion
Thus, the relationship that \( b, c, \) and \( d \) satisfy when the roots of the polynomial are in arithmetic progression is:
\[ 2b^3 + 27d = 9bc \]