What does the fundamental theorem of algebra state?

The fundamental theorem of algebra asserts that every non-constant polynomial function with complex coefficients has at least one complex root. This is a key principle in algebra that connects polynomial equations with their solutions, reflecting the nature of polynomial expressions and their behavior in the complex number system.

Understanding this theorem is crucial because it recognizes that while polynomials can have real roots, they can also have solutions that are complex numbers. In fact, the theorem guarantees that if you have a polynomial of degree n (where n is greater than zero), there will be exactly n roots in the complex number set, considering multiplicity. This means that for any polynomial equation you encounter, you can always expect at least one complex solution, even if that solution is not readily visible or appears as an imaginary number.

The other options do not accurately capture the essence of the theorem. For example, stating that all polynomial functions are linear is incorrect because polynomials can be of various degrees. The assertion that every polynomial function has a real root is not universally true, as some may have complex roots exclusively. Finally, the claim that all complex numbers are roots of polynomial functions is misleading; rather, it is about the existence of roots for a specific set of polynomials, rather than every complex number being