Talk:Polynomial remainder theorem

Latest comment: 2 years ago by 2603:7080:AFF0:60:9CB3:F538:6012:B625 in topic Generalized Polynomial Remainder Theorem

Regarding edits of 8th November 06

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Firstly, the general form for polynomial long division is:

 

and not

 

Consequently, we need to explain how to get from   to   in the second equation, which was already in place before the edits. I have now reverted this.

Secondly, "if the remainder is zero, then the linear divisor is a factor" is correct, and not "if the remainder is zero, then the linear factor is a divisor". Oli Filth 08:50, 8 November 2006 (UTC)Reply

Theorem's Discoverer?

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Okay, so how come it doesn't say who was the person that found out the Theorem? Not only it would be useful information, but doesn't the scientist deserve it? I am currently unsure of the English spelling of his name, but if nobody finds out before me, I will put it up. 79.101.183.132 (talk) 16:57, 22 May 2008 (UTC)Reply

I think it is fair to credit the Remainder Theorem to René Descartes. He certainly stated the closely related Factor Theorem explicitly in his La Géometrie in 1637 and he surely understood its relationship to polynomial division. More details can be found at the Theorem of the Day entry. —Preceding unsigned comment added by Charleswallingford (talkcontribs) 12:41, 2 November 2009 (UTC)Reply

Examples

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In the second example, in which the theorem is to be proved for the arbitrary second degree polynomal  , I added the explanation that   results from adding and subtracting arx in the numerator then regrouping terms. I believe this kind of elaboration is useful in mathematics articles, for those who understand the basic principle at work--in this case, algebraic manipulation--but still lack the experience to recognize immediately what is being done, and hence may not understand how the proof proceeds from one step to the next.Pithecanthropus4152 (talk) 21:53, 3 July 2015 (UTC)Reply

Generalized Polynomial Remainder Theorem

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Sorry, I have no editing experience on wikipedia. I would like to point out that the Remainder Theorem and 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  . Substituting   for   in   we obtain

 

which is the remainder of   on division by  . Then   does not divide   [1]. Flaudano (talk) 23:22, 28 March 2019 (UTC)Reply

References

  1. ^ Laudano, Francesco (2018). "A generalization of the remainder and factor theorem". International Journal of Mathematical Education in Science and Technology. doi:10.1080/0020739X.2018.1522676.
The first sentence of the article says essentially that the polynomial remainder theorem is a special case of Euclidean division of polynomials when the divisor has degree 1. What you propose is simply a classical way of representing Euclidean division as a rewrite rule: If the divisor is written as   where q has degree less than n, then the remainder may be eventually obtained by substituting   for   in   until all terms have a degree lower than n. This belongs to the article Euclidean division and should be added there if it is not already there. In any case, this can be generalized further to multivariate polynomials, see Multivariate division algorithm. As this way of viewing Euclidean division seems to be lacking in WP, it could be useful to add here as a generalization. D.Lazard (talk) 23:52, 28 March 2019 (UTC)Reply

The statement "... x-r is a divisor of f(x) if and only if ..." can be misleading because a divisor need not be a factor otherwise what would you call the divisor when there is a remainder? Therefore, you should change this to "...x-r is a factor of f(x)...". 2603:7080:AFF0:60:9CB3:F538:6012:B625 (talk) 14:48, 20 August 2022 (UTC)Reply