If `alpha` and `(1)/(alpha)` are zeros of `4x^(2)-17x+k-4`, then find the value of `k`.

Answer

To find the value of \( k \) given that \( \alpha \) and \( \frac{1}{\alpha} \) are the zeros of the polynomial \( 4x^2 - 17x + k - 4 \), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \( 4x^2 - 17x + (k - 4) \). Here, we can identify:
- \( a = 4 \) (coefficient of \( x^2 \))
- \( b = -17 \) (coefficient of \( x \))
- \( c = k - 4 \) (constant term)

### Step 2: Use the properties of the roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( m + n \) and the product of the roots \( m \cdot n \) can be given by:
- Sum of the roots: \( m + n = -\frac{b}{a} \)
- Product of the roots: \( m \cdot n = \frac{c}{a} \)

### Step 3: Calculate the sum of the roots
Since the roots are \( \alpha \) and \( \frac{1}{\alpha} \):
- The sum of the roots \( \alpha + \frac{1}{\alpha} = -\frac{-17}{4} = \frac{17}{4} \)

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

### Step 5: Set up the equation for the product of the roots
Using the product of the roots formula:
\[ \alpha \cdot \frac{1}{\alpha} = \frac{c}{a} \]
Substituting the values we have:
\[ 1 = \frac{k - 4}{4} \]

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

Thus, the value of \( k \) is \( 8 \).

### Summary
The value of \( k \) is \( 8 \).

---