https://file.aiquickdraw.com/mathbot/file/1728343430144kdtssp87.jpg

The given graph represents a 4th-degree polynomial, and based on the Fundamental Theorem of Algebra, we know that a polynomial of degree \(n\) has exactly \(n\) roots (real or complex). Since this is a 4th-degree polynomial, it will have exactly 4 roots in total (real or imaginary).

Now, let's analyze the graph:

- The graph crosses the x-axis at two points: near \(x = -1\) and \(x = 1\). These represent the real zeros of the polynomial.
- The total degree is 4, meaning there must be four zeros. Since the graph shows only two real zeros, the remaining two zeros must be imaginary (complex) zeros, which do not appear on the graph.

### Conclusion:
- Number of real zeros: **2**
- Number of imaginary zeros: **2**

Would you like more details or have any other questions?

Here are some related questions to further your understanding:
1. How can you factor a polynomial given its real and imaginary roots?
2. What is the significance of the multiplicity of a root in a polynomial's graph?
3. How does the behavior of a polynomial at infinity depend on its leading term?
4. What are the conditions for a polynomial to have no real roots?
5. How do complex conjugate roots appear for polynomials with real coefficients?

**Tip:** When a polynomial has complex (imaginary) roots and real coefficients, the imaginary roots always occur in conjugate pairs.

The graph of a 4th-degree polynomial is shown below. Use the Fundamental Theorem of Algebra to determine the number of real and imaginary zeros.

Learn how to use the Fundamental Theorem of Algebra to find the number of real and imaginary zeros for a 4th-degree polynomial. Analyze the graph to identify real zeros and understand how imaginary zeros occur. This guide is suitable for high school students studying polynomials and complex numbers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation