Fundamental Theorem of Algebra
A free College Algebra lesson from the “Polynomial Arithmetic and Theorems” unit, with a worked example and practice problems including step-by-step solutions.
The Fundamental Theorem of Algebra (FTA) says that every non-constant polynomial of degree n with complex coefficients has exactly n complex roots, counted with multiplicity. For polynomials with REAL coefficients, complex (non-real) roots always appear in conjugate pairs: if a + bi is a root, so is a - bi.
What you'll learn
- State the Fundamental Theorem of Algebra: every polynomial of degree n >= 1 has exactly n complex roots (counting multiplicity)
- Recognize the Complex Conjugate Root Theorem: complex roots come in conjugate pairs for real-coefficient polynomials
- Count roots correctly when there is multiplicity
Worked example
Problem. How many complex roots does the polynomial x^3 - x^2 + x - 1 have (counting multiplicity)?
- Degree is 3.
- By the Fundamental Theorem of Algebra, it has exactly 3 complex roots counting multiplicity.
Answer: 3
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. A degree-4 polynomial has how many complex roots (counting multiplicity)?
Show solution
- FTA: degree = number of complex roots.
Answer: 4
2. A degree-7 polynomial has how many complex roots (counting multiplicity)?
Show solution
- FTA.
Answer: 7
3. A degree-1 polynomial has how many complex roots?
Show solution
- A line has one root.
Answer: 1
4. P(x) = (x - 2)^3 has how many complex roots counting multiplicity?
Show solution
- x = 2 with multiplicity 3.
Answer: 3
5. P(x) = (x - 1)(x - 2)(x - 3) has how many complex roots?
Show solution
- Three distinct linear factors.
Answer: 3
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