Talk:Factor theorem

In the factor theorem for (x-a) to be a factor it is not necessary for 'a' to be a real number. 'a' can be a complex number too.

I made the Formal bit look a bit more professional, but I still don't like this article very much. 87.113.30.207 (talk) 15:15, 16 May 2008 (UTC)

stub?
Is this a stub? It seems awfully short. —Preceding unsigned comment added by 24.22.176.33 (talk) 06:01, 3 June 2008 (UTC)

Formal Version
I expanded the generality of the formal version to include the word integral domain as follows:

Let $$f$$ be a polynomial with complex coefficients, and $$a$$ be in an integral domain (e.g. $$a \in \mathbb{C}$$).

To see the proof see: http://cmes.uccs.edu/Fall2007/Math414/Videos/Math414Lecture22.mov

so we aren't restricted to $$a \in \mathbb{C}$$. I don't believe that this makes the definition too abstract.

S243a (talk) 12:01, 18 December 2010 (UTC)

Generalized Factor Theorem
Hello. I have no editing experience on wikipedia, so I apologize for the inevitable errors. I would like to point out that the factor theorem can be easily extended to non-linear divisors. Perhaps you could consider the possibility of inserting a simple example on the page, like the following.

Let $$ \ f=2x^{15} + 5x^3 +x^2-3x+3 \quad \text{and} \quad m=x^3 -2 \ $$. Substituting $$ \ 2 \ $$ for  $$ \ x^3 \  $$   in  $$ \ f(x) \  $$  we obtain

$$ \ 2\cdot2^5 + 5\cdot2 + x^2 - 3x + 3 = 64 + 10+x^2-3x+3 =x^2-3x+ 77$$

which is the remainder of $$ \ f \ $$ on division by $$ \ m$$. Then $$ \ x^3-2 \ $$ does not divide  $$ \ 2x^{15} + 5x^3 + x^2-3x +3 \ $$ .

Flaudano (talk) 23:19, 28 March 2019 (UTC)

Is this still Start class?
With a more general statement, more examples, and some proofs, does this article deserve to be Start class any more? --Svennik (talk) 18:46, 15 October 2023 (UTC)

Cluttered notation?
Can someone propose how to unclutter the notation in the introduction and elsewhere? --Svennik Svennik (talk) 18:54, 15 October 2023 (UTC)

"Commutative ring" might scare people off, so I've introduced a kind of warning
The factor theorem is often taught in schools before any ring theory. So I've tried to preface the paragraph about commutative rings with some sort of warning. I don't know whether this breaks any conventions on Wikipedia. --Svennik (talk) 12:02, 16 October 2023 (UTC)

Necessity of mentioning properties of polynomial composition
The composition operation on polynomials unsurprisingly is associative and has a unit. It's not a complete triviality to say this because polynomials are not in general functions. Any fully rigorous proof of any very basic theorem about polynomials - that uses composition of polynomials in the argument - should remark that these facts are being used. Conversely though, it's better not to clutter the main part of the argument with these considerations (which few might appreciate anyway), which is why I've decided to put this discussion in a separate note. Much of the time, polynomials are thought of as functions, and so these points are not worth making; but in general, they are not. --Svennik (talk) 16:10, 18 October 2023 (UTC)

Just to be clear, does this seem unnecessarily pedantic? I'm always uncertain where to draw the line. --Svennik (talk) 16:48, 18 October 2023 (UTC)

No warning about using advanced notions in something people learn in secondary school?
Just to be clear, we don't want to provide any warning before we unload the term "commutative ring" on the reader? --Svennik (talk) 16:45, 18 October 2023 (UTC)

Uncertainty about whether I'm in an edit war
I'm done with this page now, basically. I think any disputes are about whether my contributions follow Wikipedia conventions or not. I don't know whether I've violated 3RR. Doesn't look like it. --Svennik (talk) 16:55, 18 October 2023 (UTC)