How does the fundamental theorem of algebra describe the roots of polynomial equations?

The fundamental theorem of algebra, first proved by Carl Friedrich Gauss in 1799, provides a definitive statement regarding the solutions of polynomial equations. It establishes that every polynomial equation of degree n (where n ≥ 1) with complex number coefficients has exactly n roots in the complex number system.12

Key aspects of this theorem include:

• Existence: The theorem guarantees that every such polynomial equation has at least one root in the complex numbers.13
• Multiplicity: The n roots are not necessarily distinct. Roots may have a multiplicity greater than one, meaning a specific value can serve as a root multiple times. For example, the equation $x^2-2x+1=0$ factors to $(x-1)(x-1)=0$, where the root $x=1$ has a multiplicity of 2.14
• Scope: While the theorem ensures the existence of these roots within the complex field, it does not guarantee that these roots will exist within the specific field of the equation's coefficients (such as the rational or real numbers).4