Fundamental theorem of algebra | Wikipedia audio article
Channel: Subhajit Sahu
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra


00:01:18 1 History
00:10:54 2 Proofs
00:12:35 2.1 Complex-analytic proofs
00:29:49 2.2 Topological proofs
00:31:36 2.3 Algebraic proofs
00:34:10 2.4 Geometric proofs
00:35:00 3 Corollaries
00:35:10 4 Bounds on the zeros of a polynomial
00:37:32 5 References
00:42:13 5.1 Historic sources
00:42:25 5.2 Recent literature
00:43:03 6 External links
00:43:09 Geometric proofs
00:51:08 0.
00:51:12 Corollaries
00:52:19 Bounds on the zeros of a polynomial



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.764361664818431
Voice name: en-US-Wavenet-F


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero.
Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.
The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.
In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use some form of completeness, which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when algebra was synonymous with theory of equations.