Wherever I look on the internet I find the statement of fundamental theorem of algebra like:

Every non-constant single-variable polynomial with complex coefficients has at least one complex root.

While in school times and in the sources where the target audience are grade school students it is stated like:

Every non-zero, single-variable, degree $n$ polynomial with complex coefficients has, counted with multiplicity, exactly $n$ complex roots.

Although both of these statements are identical but I can't seem to understand the reason why it is usually presented in the manner of the former definition which appears, at least to me, a little less intuitive than the later one.

I guess the reason might be the factor of multiplicity concerned in the later definition but I would like to reflect upon what do you think about this issue.


I first thought not to cite any sources at all since the definitions are versatile on internet but upon reading the comments I changed my mind. So, FYI, I got the first definition at https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra, http://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html, https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Ld-FTA.pdf, https://brilliant.org/wiki/fundamental-theorem-of-algebra/, http://www.mathwords.com/f/fundamental_thm_algebra.htm, https://people.richland.edu/james/lecture/m116/polynomials/theorem.html and https://www.youtube.com/watch?v=shEk8sz1oOw

and second one at https://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html, http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Fund_theorem_of_algebra.html, https://www.britannica.com/topic/fundamental-theorem-of-algebra, my grade school book(NCERT book and related reference books) and most of the sources given for the first definition.

And yes, wording of the definitions I've provided are taken directly from Wikipedia.


closed as primarily opinion-based by Trevor Gunn, SchrodingersCat, Lord Shark the Unknown, Claude Leibovici, user91500 Jun 11 '17 at 8:50

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    $\begingroup$ In the first case, you can write $f(x) = (x-\alpha)\cdot g(x)$.and iterate. $\endgroup$ – Wuestenfux Jun 10 '17 at 17:09
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    $\begingroup$ The first version seems much simpler to me - why do you find the second version more intuitive? $\endgroup$ – Noah Schweber Jun 10 '17 at 17:18
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    $\begingroup$ The first formulation is entirely self-contained. The second one requires an additional definition of what the multiplicity of a root is. P.S. +1 for the question, and I don't understand the downvotes. $\endgroup$ – dxiv Jun 10 '17 at 17:23
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    $\begingroup$ I don't know what kind of answers you expect to this. This is entirely subjective and each teacher is going to present their own statement. Depending on how much time is spent on this, I can imagine both versions are taught. I also don't know why you believe it is "usually" presented one way or the other. $\endgroup$ – Trevor Gunn Jun 10 '17 at 17:28
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    $\begingroup$ I personally prefer the first one. Taken at face value, it is a simpler statement (the words involved are fewer and simpler), and I think most proofs of the second statement involves proving the first statement, then do induction on the degree, so the former statement is more fundamental as well. But ultimately it's purely a matter of preference. $\endgroup$ – Arthur Jun 10 '17 at 17:33

The issue of "multiplicity" is a sticky one - without referencing the particular structure of a polynomial, it's hard to understand what multiplicity is. In fact, when I first saw that second definition, it felt like cheating - of course every polynomial of degree $n$ has exactly $n$ roots, if you're allowed to count some of them twice! The first one leaves no sense of "cheating", and doesn't appeal to anything other than an understanding of what a "root" is.

It also reflects the construction better. The second statement suggests that the "right" way to find the roots of a polynomial is to factor it completely, and then pick out the roots. But in practice, it's much easier to pick out roots one at a time and divide out the corresponding factors. For example, you could solve $x^3 - x^2 - x + 1 = 0$ by factoring it as $(x - 1)^2(x + 1) = 0$ immediately; but it's much easier to notice that $1$ is a solution, and then use synthetic division to divide the left-hand side by $x - 1$. This is more accurately reflected in the first statement: we find just one root, and make use of it before looking for any more.

Finally, mathematicians generally prefer elegance of statement over clarity. The first statement is "slick"; the second, while possibly more useful, is clunkier. Aesthetically speaking, the first statement is preferable.


Perhaps they like the development to be parallel to that in arithmetic. First you show that every positive integer greater than one has a prime divisor. Then you show that every positive integer greater than one has a unique factorization. Math is about abstraction, and abstraction is when you see the same thing going on in two different situations. You toss out what is different and study the structure of the sameness.

So here, the pedagogues saw something similar in arithmetic and algebra and pointed it up.


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