The Fundamental Theorem of Algebra
Aligned to N-CN.C.9 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. The Fundamental Theorem of Algebra states that every polynomial of degree n has exactly n roots among the complex numbers, counting multiplicity. A quadratic (degree 2) therefore always has two roots.
Worked example
Consider x^2 + 2x + 5, which has degree 2. Solving gives x = (−2 ± √(4 − 20))/2 = (−2 ± 4i)/2 = −1 ± 2i — exactly two complex roots, just as the theorem predicts.
Practice questions
- How many roots does a degree-5 polynomial have among the complex numbers?
- Show that x^2 + 1 has exactly two roots.
- Explain why a quadratic with a negative discriminant still has two roots.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
El Teorema Fundamental del Álgebra
Este es un estándar avanzado (+). El Teorema Fundamental del Álgebra establece que todo polinomio de grado n tiene exactamente n raíces entre los números complejos, contando la multiplicidad. Por lo tanto, una cuadrática (grado 2) siempre tiene dos raíces.
Ejemplo: Considera x^2 + 2x + 5, que tiene grado 2. Al resolver se obtiene x = (−2 ± √(4 − 20))/2 = (−2 ± 4i)/2 = −1 ± 2i — exactamente dos raíces complejas, tal como predice el teorema.
More Number & Quantity lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required