Math Problem Statement
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:
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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.
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Degree of Polynomial: According to the Fundamental Theorem of Algebra, a 4th-degree polynomial will have exactly four zeros (real or imaginary).
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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:
- How can we determine the multiplicity of a zero from the graph?
- What is the difference between real and imaginary zeros in a polynomial?
- How does the degree of a polynomial relate to the maximum number of turning points?
- What is the importance of the Fundamental Theorem of Algebra in polynomial analysis?
- Can a polynomial have more imaginary zeros than real zeros?
Tip: A polynomial of degree will always have exactly zeros, though some might be real, and others may be complex (imaginary).
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Math Problem Analysis
Mathematical Concepts
Polynomials
Algebra
Graph Analysis
Formulas
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Theorems
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12
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