Information is given about a polynomial f left parenthesis x right parenthesis whose coefficients are real numbers. Find the remaining zeros of f.
Degree​ 5;  ​ zeros: 1 comma i comma negative 3 i
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Part 1
The remaining​ zero(s) of f​ is(are) 
  

​(Use a comma to separate answers as​ needed.)

The polynomial \( f(x) \) has real coefficients and is of degree 5. You are given three zeros: \( 1 \), \( i \), and \( -3i \).

Since the coefficients are real, the complex zeros must come in conjugate pairs. This means that for each complex zero, its conjugate must also be a zero of the polynomial.

- The conjugate of \( i \) is \( -i \).
- The conjugate of \( -3i \) is \( 3i \).

Thus, the remaining zeros are:

\[
- i \quad \text{and} \quad 3i
\]

So, the complete list of zeros of \( f(x) \) is:
\[
1, i, -i, -3i, 3i
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the conjugate of a complex number, and how is it used in polynomial zeros?
2. How do real coefficients affect the zeros of a polynomial?
3. Can a polynomial with real coefficients have an odd number of complex zeros?
4. What is the general form of a degree 5 polynomial given these zeros?
5. How can you factor a polynomial using its zeros?

**Tip:** When working with polynomials with real coefficients, always remember that complex zeros occur in conjugate pairs.

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 5; zeros: 1, i, -3i. The remaining zero(s) of f is(are) (Use a comma to separate answers as needed.)

Discover how to find the remaining zeros of a degree 5 polynomial given real coefficients and complex zeros. Learn about the Conjugate Root Theorem and how it applies to polynomials with real coefficients. This problem explores advanced algebraic concepts suitable for students in Grades 10-12.

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