--- title: "Roots of Polynomials and Roots of Unity" description: "A polynomial of positive degree n with complex coefficients has n complex roots counting multiplicity. For real coefficients, non-real roots occur in conjugate pairs. The nth roots of unity are evenly" 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: 2026-11-12 --- > **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 A polynomial of positive degree n with complex coefficients has n complex roots counting multiplicity. For real coefficients, non-real roots occur in conjugate pairs. The nth roots of unity are evenly spaced on 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 - Find all roots of a real cubic using a known real root, polynomial division and the remaining quadratic - 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 applies to polynomials of degree n ≥ 1 with complex coefficients. There are exactly n complex roots counting multiplicity. Real numbers are complex numbers too. Multiplicity counts repetitions: a factor (x - 8)³ counts the root 8 three times. The theorem gives a count, not the root values. It does not require the roots to be real or different. You have used a root r to find the factor x - r. Divide that factor out of a cubic, then solve the remaining quadratic. Set each factor equal to zero. For c > 0, x² + c² = 0 gives ci and -ci, since i² = -1. If all polynomial coefficients are real, every non-real root a + bi has the conjugate a - bi as a root with the same multiplicity. **Example.** For f(x) = x³ + 6x² + 25x + 150, f(-6) = 0. Divide by x + 6: subtract x²(x + 6), leaving 25x + 150, then subtract 25(x + 6), leaving 0. Thus f(x) = (x + 6)(x² + 25). Solve the factors to get -6, 5i and -5i. Over the complex numbers, f(x) = (x + 6)(x - 5i)(x + 5i). A cubic with real coefficients is continuous and has outputs of opposite signs far to the left and right. It must take the value zero between them, so it has a real root. For x³ = A with real A, take the real cube root: x³ = 27 gives x = 3. Since x³ increases, this real solution is unique. To count real roots with multiplicity, subtract the non-real root count from the degree. Let n be a positive integer. The nth roots of unity solve zⁿ = 1. Write z = r e^(iθ). Then zⁿ = rⁿ e^(inθ), so r = 1 and nθ = 2πk for an integer k. Use θ = 2πk/n for k = 0, ..., n - 1 to list each root once. Neighboring roots are 360/n degrees apart. Euler gives coordinates (cos θ, sin θ), with θ in radians: real part horizontal, imaginary part vertical. Unit-circle points (a, b) satisfy a² + b² = 1. *(drawing: The sequence shows the calculation method. The coordinates tell you where each root belongs on the Argand plane.)* **Example.** For z⁸ = 1, use θ = kπ/4 for k = 0 through 7. The eight arguments are listed below. They are 45 degrees apart, forming a regular octagon on the unit circle. At θ = π/4, cos θ = sin θ = √2/2, so the root is √2/2 + (√2/2)i and its coordinates are (√2/2, √2/2). Repeat the cosine and sine calculation for each argument. Add multiplicities to recover the degree. A factor (x + 2)² gives one distinct root, -2, counted twice. For a degree 10 polynomial, if two of its three distinct roots have multiplicities 6 and 1, the third has multiplicity 10 - 6 - 1 = 3. For a polynomial with real coefficients, a real root of even multiplicity makes the graph touch and turn at the x-axis; odd multiplicity makes it cross. **Recap.** Count roots with multiplicity, solve factors to find their values, and use equally spaced arguments to locate roots of unity on the unit circle. *The lesson, practice questions, worked solutions and flashcards were written for this dataset.* ## 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)