Math Problem Statement

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.

Solution

To determine the number of real and imaginary zeros from the given graph of a 4th-degree polynomial, we can analyze the graph's behavior:

Key observations:

  1. Real Zeros: Real zeros correspond to the points where the polynomial crosses or touches the x-axis. In this case, the graph crosses the x-axis twice, indicating two real zeros.

  2. Degree of Polynomial: According to the Fundamental Theorem of Algebra, a 4th-degree polynomial will have exactly four zeros (real or imaginary).

  3. Imaginary Zeros: Since the polynomial has two real zeros and must have four in total, the remaining zeros must be imaginary. Therefore, there are two imaginary zeros.

Conclusion:

  • Number of real zeros: 2
  • Number of imaginary zeros: 2

Would you like further explanation on how to identify zeros or additional details on the polynomial's properties?


Here are 5 related questions:

  1. How can we determine the multiplicity of a zero from the graph?
  2. What is the difference between real and imaginary zeros in a polynomial?
  3. How does the degree of a polynomial relate to the maximum number of turning points?
  4. What is the importance of the Fundamental Theorem of Algebra in polynomial analysis?
  5. Can a polynomial have more imaginary zeros than real zeros?

Tip: A polynomial of degree nn will always have exactly nn zeros, though some might be real, and others may be complex (imaginary).

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Polynomials
Algebra
Graph Analysis

Formulas

-

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12