Polynomial Roots

Fundamental Theorem of Algebra

Every non-constant polynomial p(x) with complex coefficients has at least one complex root. By repeatedly factoring out (x − r), a degree-n polynomial has exactly n complex roots (counting multiplicity).

Proven by Carl Friedrich Gauss in his 1799 PhD thesis (with topological gaps later filled). One of the deepest results in mathematics — it says ℂ is algebraically closed: every polynomial equation has a solution in ℂ.

Equivalently — every degree-n polynomial factors completely over ℂ:

p(x) = aₙ (x − r₁)(x − r₂)...(x − rₙ)

where r₁, ..., rₙ are the (not necessarily distinct) roots.

Closed-form formulas by degree

Degree	Formula	Discovered
1 (linear)	x = −b/a	Antiquity
2 (quadratic)	x = (−b ± √(b² − 4ac)) / 2a	Babylonians ~1700 BCE
3 (cubic)	Cardano's formula (uses cube roots)	Tartaglia/Cardano 1535
4 (quartic)	Ferrari's method (reduces to cubic)	Ferrari 1540
5+ (quintic and higher)	NO general formula in radicals	Abel-Ruffini 1824

Abel-Ruffini and Galois theory

The Abel-Ruffini theorem (1824) shows that no general algebraic formula (using +, −, ×, ÷, and roots √, ∛, ⁴√, ...) expresses roots of all degree-5 polynomials.

Évariste Galois (1832, killed in a duel at age 20) explained why — the obstruction lies in group theory. Each polynomial has an associated Galois group (a permutation group of its roots). The polynomial is solvable in radicals if and only if its Galois group is solvable (a technical group-theoretic condition).

Degree	Galois group	Solvable?
2	S₂ ≅ ℤ/2	Yes (cyclic)
3	S₃	Yes
4	S₄	Yes (S₄'s normal series ends in trivial)
5	S₅ (with simple A₅ inside)	NO — A₅ is simple non-abelian
≥ 5	Includes Sₙ generally	NO (in general)

This birthed group theory and abstract algebra.

Vieta's formulas — coefficients ↔ roots

For a degree-n polynomial p(x) = aₙxⁿ + a_{n-1}x^{n-1} + ... + a₀ with roots r₁, r₂, ..., rₙ:

r₁ + r₂ + ... + rₙ = −a_{n-1} / aₙ            (sum of roots)
r₁r₂ + r₁r₃ + ... + r_{n-1}rₙ = a_{n-2} / aₙ  (sum of pairwise products)
...
r₁r₂...rₙ = (−1)ⁿ a₀ / aₙ                      (product of roots)

For ax² + bx + c — sum = −b/a, product = c/a. For ax³ + bx² + cx + d — sum = −b/a, sum of pairwise products = c/a, product = −d/a.

Useful when you know coefficients and want symmetric functions of roots without solving.

Newton's method — numerical root-finding

Given f(x) and a starting guess x₀, iterate:

x_{n+1} = x_n − f(x_n) / f'(x_n)

Geometrically — at each step, draw the tangent to f at xₙ and find where it crosses the x-axis. The crossing is xₙ₊₁.

Quadratic convergence — when close to a simple root, the error squares each step. Going from 0.1 error to 0.01 to 0.0001 — doubling the correct digits each iteration.

Caveats — fails near repeated roots, can converge slowly or diverge if started too far. Newton fractals show the basins of attraction (which starting points converge to which roots) — often beautiful, with fractal boundaries.

Root multiplicity

A root r has multiplicity m if (x − r)^m divides p(x) but (x − r)^(m+1) doesn't.

Equivalent characterizations:

• p(r) = p'(r) = p''(r) = ... = p^{(m−1)}(r) = 0, but p^{(m)}(r) ≠ 0.
• The polynomial p touches the x-axis "m times" at r — but only crosses if m is odd.

For p(x) = (x − 2)³(x + 1) — root at x = 2 has multiplicity 3, root at x = −1 has multiplicity 1. Total = 3 + 1 = 4 = degree. ✓

JavaScript — finding polynomial roots

// Quadratic formula
function quadraticRoots(a, b, c) {
  const disc = b * b - 4 * a * c;
  if (disc > 0) {
    const sqrtDisc = Math.sqrt(disc);
    return [(-b + sqrtDisc) / (2 * a), (-b - sqrtDisc) / (2 * a)];
  } else if (disc === 0) {
    return [-b / (2 * a)];  // repeated root
  } else {
    // complex roots
    const realPart = -b / (2 * a);
    const imagPart = Math.sqrt(-disc) / (2 * a);
    return [`${realPart} + ${imagPart}i`, `${realPart} - ${imagPart}i`];
  }
}

console.log(quadraticRoots(1, -3, 2));   // [2, 1] — x² - 3x + 2 = (x-1)(x-2)
console.log(quadraticRoots(1, 0, 1));    // ['0 + 1i', '0 - 1i'] — x² + 1
console.log(quadraticRoots(1, -2, 1));   // [1] — (x - 1)²

// Newton's method for general polynomial
function newtonsMethod(f, fPrime, x0, maxIter = 100, tolerance = 1e-12) {
  let x = x0;
  for (let i = 0; i < maxIter; i++) {
    const fx = f(x);
    const fpx = fPrime(x);
    if (Math.abs(fpx) < 1e-15) return null;  // derivative too small
    const newX = x - fx / fpx;
    if (Math.abs(newX - x) < tolerance) return newX;
    x = newX;
  }
  return x;
}

// Example: x⁵ - x - 1 = 0 (no closed-form solution!)
const f = x => x**5 - x - 1;
const fPrime = x => 5 * x**4 - 1;

console.log(newtonsMethod(f, fPrime, 1.5));  // ~1.1673... (real root)

// Vieta's formulas for cubic ax³ + bx² + cx + d
function viestasCubic(a, b, c, d) {
  return {
    sumOfRoots: -b / a,
    sumOfProducts: c / a,
    productOfRoots: -d / a
  };
}

// x³ - 6x² + 11x - 6 = (x-1)(x-2)(x-3)
console.log(viestasCubic(1, -6, 11, -6));
// { sumOfRoots: 6, sumOfProducts: 11, productOfRoots: 6 } ✓

Where polynomial roots show up

• Algebra and pure math. Foundation of polynomial rings, field extensions, Galois theory. Algebraic numbers (roots of polynomials with integer coefficients) form a fundamental class.
• Engineering — system stability. Linear time-invariant systems are stable iff all roots of the characteristic polynomial have negative real parts. Used to analyze feedback control systems, electrical circuits, mechanical vibrations.
• Signal processing — filters. Digital filters are characterized by poles (roots of denominator polynomial) and zeros (roots of numerator). Their placement determines filter behavior.
• Numerical analysis. Many algorithms use polynomial root-finding (Newton's method, Bairstow's method, Jenkins-Traub). Critical for optimization, eigenvalue computation, ODE solvers.
• Physics. Eigenvalue problems reduce to characteristic polynomial roots. Quantum energy levels, normal modes of oscillation, all involve polynomial roots.
• Cryptography. Polynomials over finite fields and their roots underpin error-correcting codes (Reed-Solomon), elliptic-curve cryptography.
• Computer graphics — Bézier curves. Bézier curves are polynomial; root-finding is used in ray-tracing curves, intersection tests.

Common mistakes

• Forgetting complex roots exist. A degree-n polynomial with real coefficients can have fewer than n REAL roots, but always n COMPLEX roots. Always think over ℂ when counting.
• Misinterpreting multiplicity. A double root counts as 2 in the FTA total. (x − 1)²(x + 2) has degree 3 — roots are 1 (mult 2) and −2 (mult 1), total 3. Not "2 distinct roots."
• Trying to find a quintic formula. Don't waste time — Abel-Ruffini proved no general radical formula exists. For specific quintics (with special structure), formulas might exist; for general quintics, use numerical methods.
• Wrong sign in Vieta's. For aₙxⁿ + ..., sum of roots = −a_{n-1}/aₙ (not +). Always check the sign convention.
• Newton's method without checking convergence. Newton can diverge or oscillate. Always cap iterations and check convergence; otherwise infinite loop.
• Confusing roots of p with roots of p'. Roots of p are where p = 0; roots of p' are critical points of p. Repeated roots of p ARE shared with p', but that's a special case — generally roots and critical points differ.