Why do some polynomials not have closed-form zero expressions?

By Galois theory, general polynomial equations of degree 5 or higher have no formula in radicals for their roots. Their zeros exist (by the Fundamental Theorem of Algebra), but may not be expressible using a finite combination of addition, subtraction, multiplication, division, and radicals.

Zeros of Polynomials Calculator — Real & Complex Roots

Q: What is a zero of a polynomial?

A zero of a polynomial P(x) is a value a for which P(a) = 0. Equivalently, (x − a) is a factor of P(x), and a is an x-intercept of the graph of P.

Q: How many zeros does a polynomial of degree n have?

By the Fundamental Theorem of Algebra, a nonzero polynomial of degree n has exactly n complex zeros, counting multiplicity. Some may be real; the rest come in complex-conjugate pairs.

Q: What is the difference between a zero and an x-intercept?

Every real zero corresponds to an x-intercept of the graph — the point where the graph meets the x-axis at (a, 0). Complex zeros do not appear as x-intercepts on the real Cartesian plane.

Overview

This free calculator computes the real and complex zeros of a polynomial.

Definition. A value \(a\) is a zero of the polynomial \(P(x)\) if and only if \(P(a) = 0\). Equivalently, \((x - a)\) is a factor of \(P(x)\), and \(a\) is an x-intercept of the graph of \(P\) (when \(a\) is real).

Scope. The calculator uses Algebrite to find exact roots where possible. For degree ≥ 5, some polynomials have no closed-form expression for their zeros (by Galois theory), in which case the calculator may return fewer roots or a message.

Interactive Zeros Calculator

Enter the polynomial \(P(x)\) using +, -, *, /, and ^. Include the multiplication operator explicitly (e.g., 2*x^2, not 2x^2).

Input Guide & Sample Polynomials

Use standard operators: + (add), - (subtract), * (multiply), / (divide), and ^ (power). Always include the multiplication sign between a coefficient and a variable.

Sample polynomials to try:

x^2 - 9 → two real roots, \(x = \pm 3\)
x^2 + 9 → two complex roots, \(x = \pm 3i\)
x^2 + 2*x + 7 → two complex roots
x^3 + 2*x - 3 → one real root and two complex roots
3*x^4 - 3 → four complex roots (\(\pm 1\) and \(\pm i\))
x^5 + 5*x^4 + 3*x^3 + x^2 - 10*x - 120 → degree 5
x^5 + 4*x^4 - 7*x^3 - 28*x^2 + 6*x + 24 → degree 5
x^4 - 4*x^3 + 3 → degree 4 (may take a moment)

Worked Example — Find the zeros of \(P(x) = x^2 + 5x - 14\)

Step-by-step solution

Step 1 — Factor the polynomial. Find two numbers that multiply to \(ac = 1 \cdot (-14) = -14\) and add to \(b = 5\). Those numbers are \(7\) and \(-2\).

\[ P(x) = (x - 2)(x + 7) \]

Step 2 — Apply the Zero Product Property.

\[ (x - 2)(x + 7) = 0 \implies x - 2 = 0 \text{ or } x + 7 = 0 \]

Answer. The zeros are:

\[ x = 2 \quad \text{and} \quad x = -7 \]

Check: \(P(2) = 4 + 10 - 14 = 0\) ✓ and \(P(-7) = 49 - 35 - 14 = 0\) ✓

Frequently Asked Questions

What is a zero of a polynomial?

A zero of \(P(x)\) is a value \(a\) for which \(P(a) = 0\). Equivalently, \((x - a)\) is a factor of \(P(x)\), and \(a\) is an x-intercept of the graph of \(P\) (when \(a\) is real).

How many zeros does a polynomial of degree \(n\) have?

By the Fundamental Theorem of Algebra, a nonzero degree-\(n\) polynomial has exactly \(n\) complex zeros, counting multiplicity. Some may be real; the rest come in complex-conjugate pairs.

What is the difference between a zero and an x-intercept?

Every real zero corresponds to an x-intercept of the graph — the point where the graph meets the x-axis at \((a, 0)\). Complex zeros do not appear as x-intercepts on the real Cartesian plane.

Why do some polynomials have no closed-form zero expressions?

By Galois theory, general polynomial equations of degree 5 or higher have no formula in radicals for their roots. Their zeros still exist (by the Fundamental Theorem of Algebra), but they may not be expressible using a finite combination of the four arithmetic operations and radicals.

References and Related Tutorials

Sources:
[1] Algebra and Trigonometry — Swokowski, Cole — 1997 — ISBN 0-534-95308-5
[2] Algebra and Trigonometry with Analytic Geometry — Larson, Hostetler, Edwards, Heyd — 1997 — ISBN 0-669-41723-8

Last updated: October 17, 2026