If `alpha` and `beta` ar the zeros of the polynomial `f(x)=x^2-5x+k` such that `alpha-beta=1,` find the value of `kdot`

Options

A. `-5`
B. 6
C. 5
D. `-6`

Answer

To solve the problem, we need to find the value of \( k \) in the polynomial \( f(x) = x^2 - 5x + k \) given that the zeros \( \alpha \) and \( \beta \) satisfy the condition \( \alpha - \beta = 1 \).

### Step-by-Step Solution:

1. **Identify the coefficients of the polynomial**:
The polynomial is given as \( f(x) = x^2 - 5x + k \).
Here, \( a = 1 \), \( b = -5 \), and \( c = k \).

2. **Use Vieta's formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -\frac{-5}{1} = 5 \).
- The product of the roots \( \alpha \beta = \frac{c}{a} = \frac{k}{1} = k \).

3. **Set up the equations**:
We have two equations based on the information provided:
- \( \alpha + \beta = 5 \) (Equation 1)
- \( \alpha - \beta = 1 \) (Equation 2)

4. **Solve the system of equations**:
We can add Equation 1 and Equation 2:
\[ (\alpha + \beta) + (\alpha - \beta) = 5 + 1 \]
This simplifies to:
\[ 2\alpha = 6 \implies \alpha = 3 \]

5. **Find \( \beta \)**:
Now, substitute \( \alpha = 3 \) back into Equation 1:
\[ 3 + \beta = 5 \implies \beta = 5 - 3 = 2 \]

6. **Calculate the product of the roots**:
Now that we have both roots:
\[ \alpha = 3 \quad \text{and} \quad \beta = 2 \]
We can find \( k \) using the product of the roots:
\[ \alpha \beta = k \implies 3 \cdot 2 = k \implies k = 6 \]

### Final Answer:
The value of \( k \) is \( 6 \).