---
title: "Roots of Polynomials and Roots of Unity"
description: "Every polynomial of degree n has n complex roots, with non-real roots of real polynomials arriving in conjugate pairs, and the nth roots of unity sit evenly spaced around the unit circle."
canonical: https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr
source: https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Roots of Polynomials and Roots of Unity

Every polynomial of degree n has n complex roots, with non-real roots of real polynomials arriving in conjugate pairs, and the nth roots of unity sit evenly spaced around the unit circle.

Subject: Mathematics · Area: Algebra · Ages 17 to 18
Page: https://lightmysky.com/learn/mathematics/roots-of-polynomials-and-roots-of-unity-mt_XAISygqdQr

## Ready when they can

- Factor a cubic with one real root into its real root and a conjugate pair
- Find all nth roots of unity for small n and place them on the Argand plane
- State what the fundamental theorem of algebra promises and what it does not

## Lesson: Every polynomial meets its roots

The fundamental theorem of algebra promises exactly as many complex roots as the degree, counting repeats. A degree 5 polynomial owns 5, a degree 7 owns 7. It promises the count, never the values. A real cubic always hides a real root, since its odd-degree ends head opposite ways and force the graph across the axis.

**Example.** Factor first and read each piece. With (x minus 2)(x squared plus 1), the linear piece gives x = 2 and the quadratic gives x squared = minus 1, so i and minus i. The three roots are 2, i, and minus i, with the non-real pair arriving as conjugates. Likewise x cubed minus 8 has the single real root 2.

The nth roots of unity solve x to the n equals 1, giving n points spaced evenly around the unit circle. The 6th roots sit at the corners of a regular hexagon, every 60 degrees. The number i qualifies as a 4th root of unity, since i squared is minus one and squared again is 1.

**Tip.** Balance your root list against the degree before finishing. One linear factor means one real root found, and x squared plus 1 always donates the pair i and minus i. Real polynomials pair every non-real root with its conjugate, so a lone i demands a minus i nearby.

**Recap.** Degree sets the headcount, conjugates pair up, and roots of unity ring the circle evenly.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C)
- [Euler's Formula and the Exponential Form](https://lightmysky.com/learn/mathematics/eulers-formula-and-the-exponential-form-mt_RX2IxsnSwG)

## Opens up

- [Polynomial Rings, Irreducibility and Unique Factorisation](https://lightmysky.com/learn/mathematics/polynomial-rings-irreducibility-and-unique-factorisation-mt_m2G9PIHj6y)
- [Cauchy's Integral Formula and Derivatives of Every Order](https://lightmysky.com/learn/mathematics/cauchys-integral-formula-and-derivatives-of-every-order-mt_mViFwrFNWQ)