# Strictly Real Fundamental Theorem of Algebra

## TL;DR

This paper provides an accessible proof of the Fundamental Theorem of Algebra for real polynomials, demonstrating the existence of quadratic factors using only basic real analysis principles.

## Contribution

It offers a new, elementary proof of the theorem that avoids complex analysis, making it accessible to those with only fundamental real analysis knowledge.

## Key findings

- Existence of real quadratic factors for any real polynomial.
- Proof relies solely on basic real analysis concepts.
- Approachable proof suitable for educational purposes.

## Abstract

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound property for real numbers, continuity and growth property of polynomials, and unfamiliar with complex numbers, field extension or advanced topology.