Math Problem Statement
Solution
To solve this problem, we need to determine the remaining zeros of a polynomial of degree 3 with real coefficients, given that two of its zeros are and .
Key Points:
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Complex Conjugate Root Theorem: If a polynomial with real coefficients has a complex root, then the conjugate of that root must also be a root.
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Applying the Conjugate:
- Since is a zero, its conjugate must also be a zero to satisfy the requirement of real coefficients.
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Degree Check:
- The polynomial is of degree 3, so it must have exactly three roots. We already have:
- The polynomial is of degree 3, so it must have exactly three roots. We already have:
Thus, the remaining zero of is .
Answer:
The remaining zero of is:
Would you like more detailed explanations on complex roots or the Complex Conjugate Root Theorem?
Related Questions:
- How do we determine if a polynomial with real coefficients will have complex roots?
- What is the Complex Conjugate Root Theorem, and why is it important?
- Can a polynomial of degree 4 have exactly one real root?
- How do we construct a polynomial when given its roots?
- What are some real-world applications of polynomials with complex roots?
Tip: For any polynomial with real coefficients, remember that complex roots always appear in conjugate pairs.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomials
Complex Numbers
Formulas
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Theorems
Complex Conjugate Root Theorem
Suitable Grade Level
Grades 10-12
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