3.5 Complex Zeros and the Fundamental Theorem of Algebra
Ideas in this lesson
- Count degree, distinct zeros, and multiplicity
- Factor basic polynomials completely over the complex numbers
- Combine rational reduction with complex quadratic roots
- Track repeated complex zeros and complete factorization
- Use and justify conjugate pairs and multiplicities
- Construct and normalize integer polynomials from specified zeros
- Distinguish real irreducible factors from complex linear factors
- Connect real graphs to real and non-real zero counts
- Solve with non-real coefficients and test the theorem hypothesis
- Construct minimal-degree real versus complex polynomials
- Prove the odd-degree real-zero theorem and its limitations
- Find roots of unity and scaled related equations
- Construct a polynomial from a specified nonleading coefficient
- Prove impossibility using conjugate-pair degree constraints
Count degree, distinct zeros, and multiplicity · Worked example
For \(x^{3} \left(x - 4\right)^{2} \left(x^{2} + 9\right)\), state the degree, distinct zeros, each multiplicity, and total zero count over the complex numbers. Explain why the distinct count may differ from the degree. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Degree 7; \(0\) (multiplicity 3); \(4\) (multiplicity 2); \(- 3 i\); \(3 i\); total count 7.
A linear factor contributes its exponent; the quadratic contributes twice its exponent.
The quadratic zeros are \(3 i\) and \(- 3 i\) with equal multiplicities.
The listed multiplicities sum to 7. The Fundamental Theorem of Algebra counts complex zeros with multiplicity, not only distinct real intercepts.
Written-work review:
The quadratic zeros are \(3 i\) and \(- 3 i\) with equal multiplicities.
The listed multiplicities sum to 7. The Fundamental Theorem of Algebra counts complex zeros with multiplicity, not only distinct real intercepts.
Written-work review:
- A linear factor contributes its exponent; the quadratic contributes twice its exponent.
- The quadratic zeros are \(3 i\) and \(- 3 i\) with equal multiplicities.
- The listed multiplicities sum to 7. The Fundamental Theorem of Algebra counts complex zeros with multiplicity, not only distinct real intercepts.