CMClearMathAcademy

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

Why it matters: FTA is the foundation for control-systems analysis, signal processing, and any engineering or physics problem where you need to count or factor roots of a characteristic polynomial.

Worked example

Problem. How many complex roots does the polynomial x^3 - x^2 + x - 1 have (counting multiplicity)?

  1. Degree is 3.
  2. By the Fundamental Theorem of Algebra, it has exactly 3 complex roots counting multiplicity.

Answer: 3

Practice problems

1. A degree-4 polynomial has how many complex roots (counting multiplicity)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A degree-4 polynomial has how many complex roots (counting multiplicity)?
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. FTA: degree = number of complex roots.
  4. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

2. A degree-7 polynomial has how many complex roots (counting multiplicity)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A degree-7 polynomial has how many complex roots (counting multiplicity)?
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. FTA.
  4. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

3. A degree-1 polynomial has how many complex roots?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A degree-1 polynomial has how many complex roots?
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. A line has one root.
  4. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

4. P(x) = (x - 2)^3 has how many complex roots counting multiplicity?

Show solution
  1. Core Practice: First identify exactly what the question is asking: P(x) = (x - 2)^3 has how many complex roots counting multiplicity?
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. x = 2 with multiplicity 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. P(x) = (x - 1)(x - 2)(x - 3) has how many complex roots?

Show solution
  1. Core Practice: First identify exactly what the question is asking: P(x) = (x - 1)(x - 2)(x - 3) has how many complex roots?
  2. For complex numbers, use i^2 = -1 and combine real parts with real parts and imaginary parts with imaginary parts.
  3. Three distinct linear factors.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

Practice this interactively with instant feedback and an AI tutor.

Practice Fundamental Theorem of Algebra Take the free placement check

More College Algebra lessons