If one zero of the quadratic polynomial p(x) = `x^2 + 4kx – 25` is negative of the other, find the value of k.

Answer

To find the value of \( k \) in the quadratic polynomial \( p(x) = x^2 + 4kx - 25 \) given that one zero is the negative of the other, we can follow these steps:

### Step 1: Understand the relationship between the zeros
Let the zeros of the polynomial be \( \alpha \) and \( \beta \). According to the problem, if one zero is the negative of the other, we can express this as:
\[ \beta = -\alpha \]

### Step 2: Use the sum of the zeros
The sum of the zeros \( \alpha + \beta \) can be calculated using the formula for the sum of the roots of a quadratic polynomial, which is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
In our polynomial \( p(x) = x^2 + 4kx - 25 \), we have \( a = 1 \) and \( b = 4k \). Therefore:
\[ \alpha + \beta = -\frac{4k}{1} = -4k \]

### Step 3: Substitute the relationship between the zeros
Since \( \beta = -\alpha \), we can substitute this into the sum of the zeros:
\[ \alpha + (-\alpha) = 0 \]
This implies:
\[ -4k = 0 \]

### Step 4: Solve for \( k \)
From the equation \( -4k = 0 \), we can solve for \( k \):
\[ k = 0 \]

### Conclusion
The value of \( k \) is:
\[ \boxed{0} \]