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 cubic equation \( x^3 + bx^2 + cx + d = 0 \) when its roots are in arithmetic progression (AP).

### Step-by-Step Solution:

1. **Identify the Roots**:
Let the roots of the cubic equation be \( \alpha, \beta, \gamma \). Since they are in arithmetic progression, we can express them as:
\[ \alpha = \beta - d, \quad \beta = \beta, \quad \gamma = \beta + d \]
where \( d \) is the common difference.

2. **Sum of the Roots**:
The sum of the roots \( \alpha + \beta + \gamma \) can be calculated as:
\[ (\beta - d) + \beta + (\beta + d) = 3\beta \]
According to Vieta's formulas, the sum of the roots is also given by:
\[ -\frac{b}{1} = -b \]
Therefore, we can equate the two expressions:
\[ 3\beta = -b \quad \Rightarrow \quad \beta = -\frac{b}{3} \]

3. **Product of the Roots**:
The product of the roots \( \alpha \beta \gamma \) can be expressed as:
\[ (\beta - d) \cdot \beta \cdot (\beta + d) = \beta(\beta^2 - d^2) = \beta^3 - \beta d^2 \]
According to Vieta's formulas, the product of the roots is:
\[ -d \]
Thus, we have:
\[ \beta^3 - \beta d^2 = -d \]

4. **Substituting for \( \beta \)**:
Substitute \( \beta = -\frac{b}{3} \) into the product equation:
\[ \left(-\frac{b}{3}\right)^3 - \left(-\frac{b}{3}\right) d^2 = -d \]
Simplifying this gives:
\[ -\frac{b^3}{27} + \frac{b}{3} d^2 = -d \]
Rearranging leads to:
\[ \frac{b}{3} d^2 + d - \frac{b^3}{27} = 0 \]

5. **Multiplying through by 27**:
To eliminate the fractions, multiply the entire equation by 27:
\[ 9bd^2 + 27d - b^3 = 0 \]
Rearranging gives:
\[ b^3 - 9bd^2 - 27d = 0 \]

6. **Final Relation**:
This can be rearranged to the desired relation between \( b \), \( c \), and \( d \):
\[ 2b^3 + 27d = 9bc \]

### Final Answer:
Thus, the relation that \( b \), \( c \), and \( d \) satisfy is:
\[ 2b^3 + 27d = 9bc \]