https://file.aiquickdraw.com/mathbot/file/1730092958214kof68mcu.jpg

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.

Find an nth-degree polynomial function with real coefficients satisfying the given conditions: n = 4; 2i and 5i are zeros; f(−1) = 130.

Learn how to construct a 4th-degree polynomial function with real coefficients given complex roots and a specific function value. This example uses concepts like complex conjugates, difference of squares, and the Conjugate Root Theorem, making it ideal for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation