If one root of the polynomial `f(x) = x^2 + 5x + k` is reciprocal of the other, find the value of k.

Answer

To find the value of \( k \) such that one root of the polynomial \( f(x) = x^2 + 5x + k \) is the reciprocal of the other, we can follow these steps:

### Step 1: Define the Roots
Let the roots of the polynomial be \( \alpha \) and \( \frac{1}{\alpha} \).

### Step 2: Use the Sum of Roots Formula
According to Vieta's formulas, the sum of the roots of the polynomial \( ax^2 + bx + c = 0 \) is given by:
\[ \text{Sum of roots} = -\frac{b}{a} \]
For our polynomial \( f(x) = x^2 + 5x + k \), we have:
- \( a = 1 \)
- \( b = 5 \)

Thus, the sum of the roots is:
\[ \alpha + \frac{1}{\alpha} = -\frac{5}{1} = -5 \]

### Step 3: Use the Product of Roots Formula
Again, according to Vieta's formulas, the product of the roots is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
For our polynomial, this means:
\[ \alpha \cdot \frac{1}{\alpha} = \frac{k}{1} = k \]
Since \( \alpha \cdot \frac{1}{\alpha} = 1 \), we have:
\[ k = 1 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{1} \]

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