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- TL;DR Summary
- Elementary proof of FTA
I found this video showing an elementary proof of the FTA.
The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root. In other words, a polynomial of degree n will have exactly n complex roots, counting multiplicity.
The Fundamental Theorem of Algebra was first proved by mathematician Carl Friedrich Gauss in 1799. However, there have been many different proofs of this theorem since then, each using different techniques and approaches.
The Fundamental Theorem of Algebra is important because it helps us understand the behavior of polynomial equations and their roots. It is also a fundamental result in mathematics and has many applications in fields such as physics, engineering, and computer science.
One common misconception is that the theorem only applies to real numbers. However, the theorem specifically states that it applies to complex numbers. Another misconception is that the theorem guarantees the existence of real roots for polynomial equations, but this is not always the case.
Some common techniques used to prove the Fundamental Theorem of Algebra include complex analysis, topology, and linear algebra. Other techniques such as the use of Galois theory and algebraic geometry have also been used to prove this theorem.