If the sum of the product of the zeroes taken two at a time of the polynomial f(x) = `2x^(3) - 3x^(2)` + 4tx - 5 is -8, then the value of t is______.

Options

A. 2
B. 4
C. `-2`
D. `-4`

Answer

To solve the problem, we need to find the value of \( t \) in the polynomial \( f(x) = 2x^3 - 3x^2 + 4tx - 5 \) given that the sum of the product of the zeroes taken two at a time is -8.

### Step-by-step Solution:

1. **Identify the Polynomial Coefficients**:
The polynomial is given as:
\[ f(x) = 2x^3 - 3x^2 + 4tx - 5 \]
Here, the coefficients are:
- \( a = 2 \) (coefficient of \( x^3 \))
- \( b = -3 \) (coefficient of \( x^2 \))
- \( c = 4t \) (coefficient of \( x \))
- \( d = -5 \) (constant term)

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

3. **Calculate the Sum of the Product of the Roots**:
From Vieta's, we have:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{4t}{2} = 2t \]

4. **Set Up the Equation**:
We are given that the sum of the product of the zeroes taken two at a time is -8:
\[ 2t = -8 \]

5. **Solve for \( t \)**:
To find \( t \), we divide both sides by 2:
\[ t = \frac{-8}{2} = -4 \]

Thus, the value of \( t \) is \( -4 \).

### Final Answer:
The value of \( t \) is \( -4 \).