If `alpha` and `1/alpha` are the zeroes of the polynomial `4x^2-2x+(k-4)`, then value of k is:

Options

A. 4
B. 8
C. 0
D. None of these

Answer

To find the value of \( k \) for the polynomial \( 4x^2 - 2x + (k - 4) \) given that \( \alpha \) and \( \frac{1}{\alpha} \) are its zeroes, we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The polynomial is given as:
\[ 4x^2 - 2x + (k - 4) \]
Here, the coefficients are:
- \( a = 4 \)
- \( b = -2 \)
- \( c = k - 4 \)

### Step 2: Use the properties of the roots
For a quadratic polynomial \( ax^2 + bx + c \), if \( \alpha \) and \( \beta \) are the roots, then:
1. The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
2. The product of the roots \( \alpha \cdot \beta = \frac{c}{a} \)

In our case, the roots are \( \alpha \) and \( \frac{1}{\alpha} \).

### Step 3: Calculate the sum of the roots
The sum of the roots is:
\[ \alpha + \frac{1}{\alpha} = -\frac{-2}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 4: Calculate the product of the roots
The product of the roots is:
\[ \alpha \cdot \frac{1}{\alpha} = 1 = \frac{k - 4}{4} \]

### Step 5: Set up the equation for the product of the roots
From the product of the roots, we have:
\[ 1 = \frac{k - 4}{4} \]

### Step 6: Solve for \( k \)
To solve for \( k \), multiply both sides by 4:
\[ 4 = k - 4 \]
Now, add 4 to both sides:
\[ k = 4 + 4 = 8 \]

### Final Answer
The value of \( k \) is:
\[ \boxed{8} \]