If (a-b), a and (a+b) are zeroes of `2x^(3)-12x^(2)+5x-1`, then find the value of a.

Explanation

To solve the problem, we need to find the value of \( a \) given that \( (a-b) \), \( a \), and \( (a+b) \) are the roots of the polynomial \( 2x^3 - 12x^2 + 5x - 1 \).

### Step-by-Step Solution:

1. **Identify the Coefficients:**
The given polynomial is \( 2x^3 - 12x^2 + 5x - 1 \). We can identify the coefficients:
- \( a = 2 \) (coefficient of \( x^3 \))
- \( b = -12 \) (coefficient of \( x^2 \))
- \( c = 5 \) (coefficient of \( x \))
- \( d = -1 \) (constant term)

2. **Use the Relationship of Roots:**
For a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), the sum of the roots \( \alpha + \beta + \gamma \) is given by:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
Here, \( b = -12 \) and \( a = 2 \). Therefore:
\[ \alpha + \beta + \gamma = -\frac{-12}{2} = \frac{12}{2} = 6 \]

3. **Substituting the Roots:**
The roots are given as \( (a-b) \), \( a \), and \( (a+b) \). Thus:
\[ (a-b) + a + (a+b) = 6 \]

4. **Simplifying the Equation:**
Simplifying the left-hand side:
\[ (a-b) + a + (a+b) = a - b + a + a + b = 3a \]
Therefore, we have:
\[ 3a = 6 \]

5. **Solving for \( a \):**
Dividing both sides by 3 gives:
\[ a = \frac{6}{3} = 2 \]

### Final Answer:
The value of \( a \) is \( 2 \).

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