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I had asked on this forum for an easy to understand proof of the FTA.

Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients?

One of the explanations, rather than proofs that seem somewhat easier to understand has been the one given in the linked video.

https://www.youtube.com/watch?v=shEk8sz1oOw

The idea is that given a complex polynomial, for large enough input values (large in terms of absolute value of complex numbers), the output will imitate the leading term of the polynomial, i.e., the leading term will dominate for large values and the outputs will look more and more like the perfect circle we would have if we were dealing with a monomial.

Then we gradually start shrinking the input circle and by doing so the output "almost circle" also shrinks gradually, eventually intersecting the origin and proving the existence of a complex root.

​ This is not the only video that uses this technique to demonstrate why FTA must hold, but as people have mentioned in the comments, this explanation lacks certain elements that are necessary to be satisfied with the statement of FTA:

1-) It is not obvious why the almost circle that the polynomial approximates for large enough input values does not have any "holes" whereby the origin could be avoided.This I believe is a similar question to why (real) polynomial functions are continous and personally, I don't have much of a problem taking this on faith. Intuitively, it seems reasonable enough that such a curve should not have any holes/discontinuities given the principles of how a complex input is operated on in polynomials.

2-) The second point which I find more difficult to accept intuitively is the assumption that this large almost curve will "shrink continously" as we shrink the input circle. It is almost like we take a rubber band and expand it with our hands in which case no point, including the origin could escape. However how do we know that the output curve behaves in that manner?

I am certainly not looking for a rigorous proof but rather insights that may make it easier for me to wrap my mind around the two points (especially the second one) I've mentioned above.

jacob78
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  • 2-) is again continuity – user8268 Nov 20 '23 at 10:16
  • Could you edit in a link to wherever you asked for FTA proofs so we can read the comments you mentioned? I didn't see it anywhere in your recent questions. – J.G. Nov 20 '23 at 10:20
  • The comments I mentioned were those that I had seen in various videos that use this technique to explain the FTA on YouTube and elsewhere. I was not talking about the answers I've gotten on this forum. I will edit in the link to my question anyways. – jacob78 Nov 20 '23 at 10:27
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    The border moves continuously, so if a point is inside the border initially, but ends outside, at some moment it was on border. If you want more details - define function that is distance to border for points inside, distance to border multiplied by $-1$ outside - it's continuous, starts positive, ends negative. The 1st part is harder - for real case we use mean value theorem, which is not true in complex case, and continuity alone isn't enough for 1-) ($\exp$ on ${n} \times [0, 2\pi)$ is circle around $0$, but $\exp$ doesn't have zeroes). – mihaild Nov 20 '23 at 10:45
  • You seem to find continuity of polynomials more obvious in the real case than the complex one. If $f,,g$ are continuous, so are $f+g,,fg$, so you can induct on degree. – J.G. Nov 20 '23 at 11:15
  • Your underlying problem is a defective workflow. That is, you are taking a fundamentally unsound approach to learning mathematics. I recommend: [1] Seeking out a math teacher or math professor for advice. [2] Explaining to the teacher/professor exactly what your math background/education is, including those math books that you have previously successfully digested, which assumes that the book had many exercises, and that you conquered most if not all of the exercises. ...see next comment – user2661923 Nov 20 '23 at 12:33
  • [3] Explaining to the math teacher/professor where you have a gap in your education. For example, which theorem is it whose proof that you wish to understand, from the ground up. [4] Have the teacher/professor recommend a math book (with many exercises for you to solve). Open the book to page 1, and attack. Note that in general, math theory is cumulative. ...see next comment – user2661923 Nov 20 '23 at 12:37
  • This means that if a theorem is presented in the middle of a math book, you typically have to work through the book from page 1 up to the theorem, including the embedded exercises, to be confident in the validity of the proof. As far as youtube math videos (et al), these are useful tools for enjoyment and to occasionally provide insight into an isolated concept. However, the idea of using such tools to provide a solid foundation in a specific area of math is (in my opinion) an example of going off the rails. In my opinion, math is best learned from math books that contain many exercises. – user2661923 Nov 20 '23 at 12:41
  • I am aware that the way I'm approaching this may well be considered "going off the rails" since I lack a lot of prerequisite knowledge to thoroughly understand the issue in question. However that is not my actual aim at this point: I'm trying to build some intuition for the nature of polynomial functions while I keep developing my more fundamental skills, which as far as I can tell, has been quite helpful in keeping me motivated and interested in my studies. Thank you for your concern. – jacob78 Nov 20 '23 at 13:12
  • My personal experience is that as I learn the math via math books, my intuition simultaneously develops. – user2661923 Nov 21 '23 at 22:11

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