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3.5 Complex Zeros and the Fundamental Theorem of Algebra
A degree −n polynomial has exactly n complex zeros counting multiplicity. For real coefficients, every nonreal zero a+bi has a partner a−bi. Multiply conjugate factors first to get a real quadratic.
Remember Count repeated roots according to multiplicity. A complete factorization over ℂ uses linear factors.
Worked example 1
01 / State the Fundamental Theorem of Algebra, Complete Factorization Theorem, and Zeros Theorem. Then For x(x−2)2(x2+1), 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.
02 / Every nonconstant complex-coefficient polynomial has at least one complex zero. Repeated use of this fact and the Factor Theorem yields a nonzero constant times exactly as many complex linear factors as the degree. Consequently the number of complex zeros equals the degree when multiplicities are counted.
03 / A linear factor contributes its exponent; the quadratic contributes twice its exponent.
04 / The quadratic zeros are i and −i with equal multiplicities.
05 / The listed multiplicities sum to 5. The Fundamental Theorem of Algebra counts complex zeros with multiplicity, not only distinct real intercepts.
Worked example 2 · Count degree, distinct zeros, and multiplicity
For x2(x−3)3(x2+4)2, 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.
A linear factor contributes its exponent; the quadratic contributes twice its exponent. The quadratic zeros are 2i and −2i with equal multiplicities. The listed multiplicities sum to 9. 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 2i and −2i with equal multiplicities.
The listed multiplicities sum to 9. The Fundamental Theorem of Algebra counts complex zeros with multiplicity, not only distinct real intercepts.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.