Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

3: 3. The fundamental theorem of algebra and factoring polynomials

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    The similarities and differences between \(\mathbb{R}\) and \(\mathbb{C}\) can be described as elegant and intriguing, but why are complex numbers important? One possible answer to this question is the Fundamental Theorem of Algebra. It states that every polynomial equation in one variable with complex coefficients has at least one complex solution. In other words, polynomial equations formed over \(\mathbb{C}\) can always be solved over \(\mathbb{C}\). This amazing result has several equivalent formulations in addition to a myriad of different proofs, one of the first of which was given by the eminent mathematician Carl Gauss in his doctoral thesis.