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The question provides a polynomial \( p(x) \) with known zeros at \( x = \frac{1}{5}, x = -4, \) and \( x = 2 \). The task is to determine the correct form of the polynomial based on these zeros.

### Step-by-step process:

1. **Zeros of the Polynomial:**
   - For \( x = \frac{1}{5} \), the factor will be \( (5x - 1) \) (as setting \( 5x - 1 = 0 \) leads to \( x = \frac{1}{5} \)).
   - For \( x = -4 \), the factor will be \( (x + 4) \).
   - For \( x = 2 \), the factor will be \( (x - 2) \).

2. **Form of the Polynomial:**
   A polynomial is formed by multiplying the factors corresponding to the zeros. Thus, the polynomial can be written as:
   \[
   p(x) = (5x - 1)(x + 4)(x - 2)
   \]
   This corresponds to option **D**.

### Correct Answer:
The correct equation of the polynomial is **Option D: \( p(x) = (5x - 1)(x + 4)(x - 2) \)**.

Would you like further details on how to expand or verify this? Here are some follow-up questions to deepen understanding:

1. How can we expand \( (5x - 1)(x + 4)(x - 2) \) to find the expanded polynomial form?
2. How do we determine the degree of the polynomial based on its factors?
3. Why does each zero correspond to a factor in the polynomial?
4. How would the polynomial change if one of the zeros was a double root?
5. What does the sign of each factor tell us about the behavior of the polynomial around its zeros?

**Tip:** The degree of a polynomial is the sum of the powers of its factors—each linear factor contributes a power of 1.

A polynomial p has zeros when x = 1/5, x = -4, and x = 2. What could be the equation of p?

This problem explores how to find the equation of a polynomial that has zeros at x = 1/5, x = -4, and x = 2. The step-by-step solution includes factoring and applying the Zero-Factor Theorem. Suitable for students in Grades 9-12.

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