Let's analyze and solve this problem step-by-step.

1. Given Information:

   • Degree of polynomial $n = 3$
   • Zeros of the polynomial: $2$ and $4i$
   • $f(-1) = 51$

2. Understanding the Polynomial: Since the polynomial has complex roots, we need to consider that complex roots come in conjugate pairs (i.e., if $4i$ is a root, then $-4i$ must also be a root). This gives us the roots:

   • $2$
   • $4i$
   • $-4i$

3. Constructing the Polynomial: The polynomial can be represented as: $f(x) = a(x - 2)(x - 4i)(x + 4i)$ where $a$ is a constant.

4. Simplifying the Factors: We simplify $(x - 4i)(x + 4i)$ using the difference of squares: $(x - 4i)(x + 4i) = x^2 - (4i)^2 = x^2 - 16$ Therefore, the polynomial becomes: $f(x) = a(x - 2)(x^2 - 16)$

5. Using $f(-1) = 51$ to Find $a$: Substitute $x = -1$ and set $f(-1) = 51$: $f(-1) = a(-1 - 2)((-1)^2 - 16)$ $f(-1) = a(-3)(1 - 16)$ $f(-1) = a(-3)(-15)$ $f(-1) = 45a$ Since $f(-1) = 51$, we have: $45a = 51$ $a = \frac{51}{45} = \frac{17}{15}$

6. Final Polynomial: Substitute $a = \frac{17}{15}$ back into the polynomial: $f(x) = \frac{17}{15}(x - 2)(x^2 - 16)$

Thus, the polynomial $f(x)$ is: $f(x) = \frac{17}{15}(x - 2)(x^2 - 16)$

Would you like a detailed explanation of each step or have any questions?

Here are some related questions to explore further:

1. How would you verify that the roots indeed satisfy this polynomial?
2. What would happen if you changed one of the roots? How would that affect the polynomial?
3. How can you use synthetic division to confirm that these roots work?
4. What does it mean for a polynomial to have complex roots, and how does it affect its symmetry?
5. How would the polynomial change if the constant $a$ were set to a different value?

Tip: When constructing polynomials from roots, always remember to include conjugates for complex roots to ensure the polynomial has real coefficients.