What does the Fundamental Theorem of Algebra state?

• A. All polynomials have only real roots
• B. Every non-zero polynomial equation has at least one complex root
• C. All polynomial functions are continuous
• D. Every polynomial can be factored into linear terms

Answer: (B)

The Fundamental Theorem of Algebra asserts that every non-zero polynomial equation of degree n, where n is greater than zero, has at least one complex root. This is a foundational principle in algebra, affirming that polynomial equations are anchored in the complex number system. For instance, a polynomial of degree five will have at least one root, which may be real or complex. This theorem showcases the completeness of the complex number system in relation to polynomials.

The significance of this theorem extends beyond simply stating that roots exist; it also underlines the interaction between algebra and complex analysis. This means that even if all roots are not real numbers, the theorem guarantees their existence within the larger framework of complex numbers.

In contrast, the other statements either misrepresent the properties of polynomials or do not accurately encapsulate the core assertion of the theorem. For example, while it is true that all polynomial functions are continuous and that every polynomial can be factored into linear terms over the complex numbers, these statements do not encapsulate the essential idea of guaranteeing the presence of at least one root, whether real or complex.