Polynomial Equations with Complex Roots

The Fundamental Theorem of Algebra

One of the most celebrated results in all of mathematics is the Fundamental Theorem of Algebra: every polynomial of degree n (with complex, and in particular, real coefficients) has exactly n roots when counted with multiplicity in the complex number system ℂ. This theorem was first proved rigorously by Gauss in his 1799 doctoral thesis. It tells us that the complex numbers are “algebraically closed” — you can never escape ℂ by solving polynomial equations.

The Conjugate Root Theorem

For polynomials with real coefficients, complex roots always come in conjugate pairs. If z = a + bi (with b ≠ 0) is a root, then so is z̅ = a − bi. The proof is elegant: if P(z) = 0 and all coefficients are real, then taking the conjugate of both sides gives P(z̅) = 0̅ = 0 (since conjugation distributes over addition and multiplication, and conjugating a real number leaves it unchanged).

This theorem is enormously practical: once you find one complex root of a real polynomial, you immediately know another root — its conjugate. For a cubic with real coefficients, there must be either three real roots, or one real root and one conjugate pair.

Building Real Quadratic Factors

Given a complex conjugate pair z = a + bi and z̅ = a − bi, the corresponding quadratic factor is:

(x − z)(x − z̅) = x² − (z + z̅)x + z z̅ = x² − 2ax + (a² + b²)

This quadratic has real coefficients and is irreducible over ℝ (it cannot be factored into real linear factors). Memorising the structure x² − 2ax + (a²+b²) saves time — the middle coefficient is minus twice the real part, and the constant term is the squared modulus.

Strategy for Higher-Degree Polynomials

To factor a cubic or quartic with real coefficients when one complex root is known:

Step 1: Write down the conjugate root as well (conjugate root theorem).

Step 2: Form the real quadratic factor (x² − 2ax + (a²+b²)).

Step 3: Divide the polynomial by this quadratic factor using polynomial long division. The quotient will have real coefficients.

Step 4: Solve the quotient equation to find remaining roots.

Solving zⁿ = c Over the Complex Numbers

Equations of the form zⁿ = c (where c is a real or complex number) are solved using polar form. Write c = r cisθ, then z = r^1/n cis((θ + 2kπ)/n) for k = 0, 1, …, n−1. This gives n equally spaced roots on a circle of radius r^1/n. When c is a positive real number, the roots include one real root (k=0) and the rest are complex, coming in conjugate pairs.

Connecting to the Fundamental Theorem

Once we know all n roots z&sub1;, z&sub2;, …, z_n of a degree-n polynomial P(x), we can write P(x) = a(x − z&sub1;)(x − z&sub2;) … (x − z_n) where a is the leading coefficient. For polynomials with real coefficients, complex roots pair up, and we group each pair into a real quadratic factor to express P(x) as a product of real linear and real quadratic factors. This factored form over ℝ is important for integration, partial fractions, and graphical analysis.