Consider biquadratic equation `81x^(4) + 216x^(3) +96x = 65`, whose roots are `a,b,c,d`. Given `a,b`, real roots and c,d are imaginary roots.
On the basis of above information, answer the followin questions:
The value of `(a+b)^(3) - (c+d)^(3)` is equal to-

Options

A. `(-128)/(3)`
B. 0
C. `(-64)/(3)`
D. `(-142)/(9)`

Explanation

To solve the biquadratic equation \( 81x^4 + 216x^3 + 96x = 65 \) and find the value of \( (a+b)^3 - (c+d)^3 \), where \( a, b \) are real roots and \( c, d \) are imaginary roots, we will follow these steps:

### Step 1: Rearrange the Equation
First, we rearrange the equation to set it to zero:
\[ 81x^4 + 216x^3 + 96x - 65 = 0 \]

### Step 2: Factor the Equation
We can factor the left-hand side. Notice that we can rewrite the equation:
\[ 81x^4 + 216x^3 + 96x = 65 \]
This can be factored as:
\[ (3x + 2)^4 - 9^2 = 0 \]
This is in the form of \( a^2 - b^2 = 0 \), which can be factored as:
\[ (3x + 2)^2 - 9 = 0 \]

### Step 3: Solve for Real Roots
Now, we solve the two equations:
1. \( (3x + 2)^2 - 9 = 0 \)
2. \( (3x + 2)^2 + 9 = 0 \)

From the first equation:
\[ (3x + 2)^2 = 9 \]
Taking the square root gives:
\[ 3x + 2 = 3 \quad \text{or} \quad 3x + 2 = -3 \]
Solving these:
1. \( 3x + 2 = 3 \) leads to \( 3x = 1 \) or \( x = \frac{1}{3} \)
2. \( 3x + 2 = -3 \) leads to \( 3x = -5 \) or \( x = -\frac{5}{3} \)

Thus, the real roots are \( a = \frac{1}{3} \) and \( b = -\frac{5}{3} \).

### Step 4: Solve for Imaginary Roots
From the second equation:
\[ (3x + 2)^2 + 9 = 0 \]
This gives:
\[ (3x + 2)^2 = -9 \]
Taking the square root gives:
\[ 3x + 2 = 3i \quad \text{or} \quad 3x + 2 = -3i \]
Solving these:
1. \( 3x + 2 = 3i \) leads to \( 3x = 3i - 2 \) or \( x = -\frac{2}{3} + i \)
2. \( 3x + 2 = -3i \) leads to \( 3x = -3i - 2 \) or \( x = -\frac{2}{3} - i \)

Thus, the imaginary roots are \( c = -\frac{2}{3} + i \) and \( d = -\frac{2}{3} - i \).

### Step 5: Calculate \( a + b \) and \( c + d \)
Now, we calculate:
\[ a + b = \frac{1}{3} - \frac{5}{3} = -\frac{4}{3} \]
\[ c + d = \left(-\frac{2}{3} + i\right) + \left(-\frac{2}{3} - i\right) = -\frac{4}{3} \]

### Step 6: Calculate \( (a+b)^3 - (c+d)^3 \)
Now we substitute into the expression:
\[ (a+b)^3 - (c+d)^3 = \left(-\frac{4}{3}\right)^3 - \left(-\frac{4}{3}\right)^3 \]
Since both terms are equal, we have:
\[ (a+b)^3 - (c+d)^3 = 0 \]

### Final Answer
Thus, the value of \( (a+b)^3 - (c+d)^3 \) is:
\[ \boxed{0} \]