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I am currently in AP Calculus BC and one more year to go, I have heard about Fundamental Theorem of Algebra several times, and with the resources that is out there today I tried to search and study myself. (Also noticed there are some inconsistencies in the versions of explanation about this theorem). But I still wonder why they don't teach this in High School in the USA? Am I wrong? Or do they teach this in other countries?

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    @HWinslow: You might find the following worth looking at: *Appendix: Cauchy's proof that every equation has a root* on pp. 577-581 of **Advanced Course in Algebra** by Webster Wells (1904) http://books.google.com/books/reader?id=aOM5AAAAMAAJ2012-03-21

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You may need complex analysis or topology to prove it,so this is for advanced undergraduate(junior, senior) , see http://books.google.com/books?id=g0KHD7EIl4cC&printsec=frontcover&dq=fundamental+theorem+of+algebra&hl=en&sa=X&ei=ASNpT-O1B4eMgwfJn4ydCQ&ved=0CDsQ6AEwAA#v=onepage&q=fundamental%20theorem%20of%20algebra&f=false

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    Actually, there's a proof that uses only elementary analysis. See, for example, Volume 1 of Mathematical Analysis by Zorich, p 283-284.2019-05-24