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.