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Sometimes a theorem is so important that it becomes known as a fundamental theorem in mathematics. This is the case for the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that every polynomial equation of degree n, greater than or equal to one, has exactly n complex zeros. In fact, there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. The Fundamental Theorem of Algebra can also tell us when we have factored a polynomial completely but does not tell us how to factor a polynomial completely. Carl Friedrich Gauss was the first person to completely prove the theorem. In this paper, we will explore the theorem in more depth.

Early studies of equations by al'Khwarizmi, an Islamic mathematician, only allowed positive real roots and so the Fundamental Theorem of Algebra was not relevant. It wasn’t until the 1500’s that an Italian mathematician by the name of Cardan was able to realize that one could work with “complex numbers” rather than just the real numbers. This discovery was made in the course of studying a formula, which gave the roots of a cubic equation. Bombelli, in his book Algebra, published in 1572, produced a set of rules for manipulating these "complex numbers”. In 1637, Descartes, the father of analytic geometry, observed that one could imagine for every equation of degree n, you would get n roots but these roots would not be a real number.

In 1629, a Flemish mathematician, Albert Girard, published a book called L’invention nouvelle en l’ Algebre. In his book, he claimed that there were always n solutions for equations of degree n. However he did not assert that solutions are of the form a + bi, w...

... middle of paper ...

...eneral theorem on the existence of a minimum of a continuous function.

Two years after Argand's proof, Gauss published a second proof of the Fundamental Theorem of Algebra. Gauss used Euler's approach but instead of operating with roots, Gauss operated with indeterminates. This proof was considered complete and correct. A third proof was written by Gauss and like the first, was topological in nature. In 1849 Gauss produced his 4th proof of the Fundamental Theorem of Algebra. This was different in that he proved that a polynomial equation of degree n with complex coefficients has n complex roots.

As you can see, the path to proving the Fundamental Theorem of Algebra started way back. Many mathematicians tried and progress was gradually made. Eventually, Gauss was the first to produce a complete theorem although it is thought that Gauss finished what Euler started.

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