The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex coefficients has at least one solution within the complex numbers. This means that for any polynomial like \( ax^n + \dots + k = 0 \), there is at least one value of \( x \) (which could be a complex number involving real and imaginary parts) that satisfies the equation. As a result, complex numbers are essential to fully solving polynomial equations, ensuring no solutions are "missing" when compared to real numbers alone. This theorem underpins much of modern algebra and ensures the completeness of solutions in the complex plane.