If one zero of `p(x)=4x^(2)-(8k^(2)-40k)x-9` is negative of the other, find values of `k`.

Answer

To solve the problem, we need to find the values of \( k \) such that one zero of the polynomial \( p(x) = 4x^2 - (8k^2 - 40k)x - 9 \) is the negative of the other zero.

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the roots of the polynomial be \( m \) and \( -m \).

2. **Using Vieta's Formulas**:
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( m + (-m) = 0 \) (which is satisfied here).
- The product of the roots \( m \cdot (-m) = -m^2 \).

3. **Setting Up the Polynomial**:
The polynomial can be expressed as:
\[ p(x) = 4x^2 - (8k^2 - 40k)x - 9 \]
Here, \( a = 4 \), \( b = -(8k^2 - 40k) \), and \( c = -9 \).

4. **Applying the Factor Theorem**:
Since \( m \) is a root, we can substitute \( x = m \) into the polynomial:
\[ p(m) = 4m^2 - (8k^2 - 40k)m - 9 = 0 \]
This gives us our first equation.

5. **Substituting the Negative Root**:
Now, substitute \( x = -m \):
\[ p(-m) = 4(-m)^2 - (8k^2 - 40k)(-m) - 9 = 0 \]
Simplifying this:
\[ 4m^2 + (8k^2 - 40k)m - 9 = 0 \]
This gives us our second equation.

6. **Setting the Two Equations Equal**:
Since both \( p(m) = 0 \) and \( p(-m) = 0 \), we can equate the two expressions derived from substituting \( m \) and \( -m \):
\[ 4m^2 - (8k^2 - 40k)m - 9 = 4m^2 + (8k^2 - 40k)m - 9 \]
Simplifying this, we eliminate \( 4m^2 \) and \( -9 \) from both sides:
\[ -(8k^2 - 40k)m = (8k^2 - 40k)m \]
This leads to:
\[ -2(8k^2 - 40k)m = 0 \]

7. **Factoring Out**:
Since \( m \neq 0 \) (as \( m \) is a root), we can factor out:
\[ 8k^2 - 40k = 0 \]

8. **Solving the Quadratic Equation**:
Factoring gives:
\[ 8k(k - 5) = 0 \]
Setting each factor to zero gives:
\[ 8k = 0 \quad \Rightarrow \quad k = 0 \]
\[ k - 5 = 0 \quad \Rightarrow \quad k = 5 \]

### Final Values of \( k \):
Thus, the values of \( k \) are:
\[ k = 0 \quad \text{and} \quad k = 5 \]