Lesson 45: Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-constant polynomial function of degree n ≥ 1 has exactly n complex roots (counting multiplicities). This theorem guarantees that polynomial equations have solutions in the complex number system.
Explanation
- Polynomial of degree 1 → exactly 1 root
- Polynomial of degree 2 → exactly 2 roots (may be real or complex)
- Polynomial of degree 3 → exactly 3 roots, etc.
- Some roots may be repeated (multiplicity > 1)
- Real coefficients imply complex roots occur in conjugate pairs
Examples
Example 1: f(x) = x³ − 6x² + 11x − 6
- Degree 3 → 3 roots
- Factor: (x − 1)(x − 2)(x − 3)
- Zeros: x = 1, 2, 3
Example 2: f(x) = x² + 1
- Degree 2 → 2 roots
- Equation: x² + 1 = 0 → x² = −1 → x = i, −i
- Zeros are complex conjugates
Applications
The theorem is used to:
- Determine the total number of roots of a polynomial
- Understand the structure of complex roots
- Confirm that factoring and synthetic division account for all roots
- Predict graph behavior for polynomials of higher degrees
Practice Problems
- State the number of roots for f(x) = 2xⴠ− 3x³ + x − 5
- Find all roots (real and complex) of f(x) = x² + 4
- Factor f(x) = x³ − 3x² + 4x − 12 and identify all roots
- Verify that a polynomial of degree 5 has 5 roots counting multiplicity
- Explain why f(x) = xâ´ + 1 has 4 complex roots