Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra states that every non-constant polynomial with complex (including real) coefficients has at least one complex root, and consequently a polynomial of degree n has exactly n roots counted with multiplicity in the complex number system. This theorem guarantees that the complex numbers are algebraically closed, meaning no further number extensions are needed to solve polynomial equations. It was first proved rigorously by Carl Friedrich Gauss in his 1799 doctoral dissertation and is one of the most important results in all of mathematics.
Key Formula
p(x) = a_n * x^n + ... + a_0 = a_n*(x−r1)(x−r2)···(x−rn)
LaTeX: p(x) = a_n x^n + \cdots + a_1 x + a_0 = a_n(x - r_1)(x - r_2)\cdots(x - r_n)
| Symbol | Meaning | Unit |
|---|---|---|
| p(x) | polynomial of degree n with complex coefficients | dimensionless |
| n | degree of the polynomial | dimensionless |
| aₙ | leading coefficient (aₙ ≠ 0) | dimensionless |
| r₁, r₂, …, rₙ | n roots (possibly complex, counted with multiplicity) | dimensionless |
Worked Example
Problem
Given p(x) = x⁴ − 1, find all four roots guaranteed by the Fundamental Theorem of Algebra.
Solution
Step 1: Factor as difference of squares: x⁴ − 1 = (x² − 1)(x² + 1). Step 2: Factor further: (x² − 1) = (x − 1)(x + 1). Step 3: Solve x² + 1 = 0 → x² = −1 → x = ±i. Step 4: All four roots: x = 1, x = −1, x = i, x = −i. Step 5: Verify: degree is 4, and we found exactly 4 roots (2 real, 2 complex conjugate), confirming the theorem.
Answer
Roots: x = 1, −1, i, −i (2 real and 2 purely imaginary, totaling 4 roots for degree 4)
Consequences and Corollaries of the Fundamental Theorem of Algebra
| Statement | Implication | Example | Notes |
|---|---|---|---|
| Degree n → n roots | Every poly. of degree n has exactly n complex roots | x³ has 3 roots | Counted with multiplicity |
| Complex conjugate roots | Real-coeff polys have conjugate pairs | x²+1 → ±i | Non-real roots come in pairs |
| Factorisation | p(x) = aₙ(x−r₁)···(x−rₙ) | x²−5x+6=(x−2)(x−3) | Complete factorisation over ℂ |
| Real factorisation | Real polys factor into linears & irreducible quadratics | x³−1=(x−1)(x²+x+1) | Over reals only |
| Algebraic closure | ℂ is algebraically closed | No poly. over ℂ lacks a root | Key property of complex numbers |
| Odd-degree polys | At least one real root | x³+x+1=0 has one real root | Complex conjugates come in pairs |
Interactive Tools
Khan Academy — Fundamental Theorem of Algebra
Conceptual lessons and worked examples illustrating the theorem and its applications.
Open Tool
Wikimedia Commons, CC BY-SA
Related Terms
Complex Number
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit satisfying i² = −1. The real part a and imaginary part b together extend the real number line into a two-dimensional complex plane, enabling solutions to equations like x² + 1 = 0 that have no real solutions. Complex numbers are fundamental in electrical engineering, quantum mechanics, signal processing, and control theory.
Descartes' Rule of Signs
Descartes' Rule of Signs states that the number of positive real roots of a polynomial p(x) with real coefficients is either equal to the number of sign changes between consecutive nonzero coefficients, or is less than that number by an even integer. Similarly, the number of negative real roots equals the number of sign changes in p(−x), or differs by a positive even integer. This rule provides an upper bound on the number of positive and negative real roots without actually solving the polynomial.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
The word "fundamental" comes from the Latin "fundamentalis," meaning "of the foundation." The theorem is called fundamental because it underpins the entire theory of polynomial equations and algebraic number theory. Carl Friedrich Gauss gave the first rigorous proof in his 1799 doctoral dissertation at the University of Helmstedt, with three additional proofs following over his career. Earlier partial results were contributed by Peter Roth (1608), Jean le Rond d'Alembert (1746), and Leonhard Euler.