What is the Fundamental Theorem of Algebra?
Polynomial Equations
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The Fundamental Theorem of Algebra (FTA) is nothing new; it was first proved by mathematician Carl Friedrich Gauss (1777–1855) in 1799. The equation was as follows:
The proof of this theorem goes on for pages—far beyond the scope of this book. What all those proofs, numbers, and letters boil down to is that a polynomial equation must have at least one number in its solution. It also tells us when we have factored a polynomial completely. Simple enough, but like much of mathematics, someone had to prove it.
But that is not all: The FTA is not constructive, and therefore it does not tell us how to completely factor a polynomial. In other words, in reality, no one really knows how to factor a polynomial, we only know how to apply techniques to certain kinds of polynomials. In fact, French mathematician Evariste Galois (1811–1832), who died tragically in a duel, proved that there will never be a general formula that will solve fifth degree or higher polynomials.