To find an nth-degree polynomial function $f(x)$ with real coefficients that satisfies the conditions provided:

1. Given Parameters:

   • $n = 4$: This means the polynomial is of degree 4.
   • Zeros: $2i$ and $5i$.
   • $f(-1) = 130$.

2. Understanding Complex Roots: Since the polynomial has real coefficients, the complex roots must come in conjugate pairs. Therefore, the roots of the polynomial are:

   • $2i$ and $-2i$,
   • $5i$ and $-5i$.

3. Constructing the Polynomial: Using the roots, we can write the polynomial in terms of its factors: $f(x) = a(x - 2i)(x + 2i)(x - 5i)(x + 5i)$ Each pair of factors can be simplified using the difference of squares:

   • $(x - 2i)(x + 2i) = x^2 + 4$,
   • $(x - 5i)(x + 5i) = x^2 + 25$.

   So, the polynomial becomes: $f(x) = a(x^2 + 4)(x^2 + 25)$

4. Expanding the Polynomial: Now, expand $(x^2 + 4)(x^2 + 25)$: $f(x) = a(x^4 + 25x^2 + 4x^2 + 100)$ $f(x) = a(x^4 + 29x^2 + 100)$

5. Determining the Value of $a$: We use the condition $f(-1) = 130$ to find $a$: $f(-1) = a((-1)^4 + 29(-1)^2 + 100)$ $f(-1) = a(1 + 29 + 100)$ $f(-1) = a \cdot 130$ Since $f(-1) = 130$, we have: $a \cdot 130 = 130$ $a = 1$

6. Final Polynomial: Substitute $a = 1$ into the polynomial: $f(x) = x^4 + 29x^2 + 100$

Answer:

The polynomial function is: $f(x) = x^4 + 29x^2 + 100$

Would you like more details, or do you have any questions?

Here are 5 questions to extend your understanding:

1. How do complex roots affect the form of a polynomial with real coefficients?
2. Why do complex roots appear in conjugate pairs in polynomials with real coefficients?
3. How can you verify if a polynomial function satisfies a given point, such as $f(-1) = 130$?
4. What other methods can be used to construct a polynomial given certain roots and conditions?
5. How does the value of $a$ affect the shape and position of the polynomial graph?

Tip: When given complex roots, always check if real coefficients are required; this will guide you to add the conjugate roots automatically.