Suppose that a polynomial function of degree 5 with rational coefficients has the given numbers as zeros: -5, 0, 2, -i, 4. Find the other zeros.

To determine the remaining zeros of the polynomial, we can use the fact that a polynomial with rational coefficients must have complex roots appear in conjugate pairs. Here’s the breakdown:

### Given Zeros:
- \( -5 \)
- \( 0 \)
- \( 2 \)
- \( -i \)
- \( 4 \)

Since the polynomial has rational coefficients, any complex roots must occur in conjugate pairs. Therefore, if \( -i \) is a root, its conjugate \( i \) must also be a root.

### List of Zeros:
Thus, the complete list of zeros for this polynomial is:
- \( -5 \)
- \( 0 \)
- \( 2 \)
- \( -i \)
- \( 4 \)
- \( i \)

The other zero, therefore, is \( i \).

Would you like further details on constructing the polynomial from these roots, or have any other questions?

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Here are 5 questions to explore further:

1. How do you construct the polynomial using these zeros?
2. Why must complex roots appear in conjugate pairs for polynomials with rational coefficients?
3. How do you determine the degree of the polynomial given its roots?
4. What happens if a polynomial has irrational roots along with rational coefficients?
5. How can we verify that all roots have been accounted for in a polynomial?

**Tip:** When dealing with polynomials with rational coefficients, always check if complex or irrational roots need to appear in conjugate pairs for the polynomial to remain valid.

Explore how to find the missing zero of a fifth-degree polynomial with rational coefficients when given zeros include both real and complex values. This solution covers complex conjugates and the Conjugate Root Theorem, ideal for high school algebra students.

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