What are the factors of these two cubic polynomial expressions: 2x+x-27x-36 and 7x-13x+5x+1? Although their factors give real zeros, generally speaking, zeros of polynomial of second degree or higher was proved by Gauss in 1799 to extend into the field of complex numbers. This mathematical assertion is known as the fundamental theorem of algebra. Gauss gave several proofs. However, none is rigorous by modern standards. So called modern rigorous but non-elegant proofs require the use of analysis and therefore are not really algebraic. In a sense, Gauss proofs were based on geometric considerations of constructability while modern analytic approach relies on the method of the calculus and its connection to the concept of infinitesimal and sum of infinitesimals by infinite series expansions of functions with notions of convergence and the existence of a limit to a functional sequence.
Unfortunately, these methodologies of the calculus makes analysis equivalent to the mathematics of approximation. This rigid reliance on an approximate approach to reach mathematical truth and certainty imposes a heavy burden on any theory of proof for the acceptance of analysis as the ultimate branch of modern mathematics, for example, the 200-page proof of Fermat’s last theorem and the equally lengthy proof for the existence of asymptotic freedom in the standard model of elementary particles.
Re: algebraic factors -
02-25-2008, 03:26 PM
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