Briot-Ruffini method: known only in Brazil?
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From: Bráulio Bezerra on 13 Sep 2009 12:16 Hi, Virtually every kid in Brazil learns to divide polynomials by one of the form (x - a) using the Briot-Ruffini method. It is just a long division method, but works a little faster since someone writes only the coefficients. It also has numerical applications, since it can be used to find the value of a polynomial and its derivatives of all orders at some given point, in a particularly efficient way. This is done viewing the polynomial as (((a_4x + a_3) x + a_2) x + a_1 ) x + a_0, for example. The problem is that this method is only mentioned by Brazilian people in books, on the Internet, everywhere. So I don't know if this method is so obvious that it commonly deserves no mention; if this is just a trend we have in Brazilian education; or if it has another name outside Brazil. Any thoughts, anyone?
From: Bráulio Bezerra on 13 Sep 2009 12:24 Replying to myself, but not completely. I've found this http://en.wikipedia.org/wiki/Ruffini%27s_rule. This is a start, at least. There is also http://en.wikipedia.org/wiki/Horner_scheme, which deals with derivatives of polynomials. But there's no mention of someone called Briot. On Sep 13, 1:16 pm, Bráulio Bezerra <brauliobeze...(a)gmail.com> wrote: > Hi, > > Virtually every kid in Brazil learns to divide polynomials by one of > the form (x - a) using the Briot-Ruffini method. It is just a long > division method, but works a little faster since someone writes only > the coefficients. It also has numerical applications, since it can be > used to find the value of a polynomial and its derivatives of all > orders at some given point, in a particularly efficient way. This is > done viewing the polynomial as (((a_4x + a_3) x + a_2) x + a_1 ) x + > a_0, for example. > > The problem is that this method is only mentioned by Brazilian people > in books, on the Internet, everywhere. So I don't know if this method > is so obvious that it commonly deserves no mention; if this is just a > trend we have in Brazilian education; or if it has another name > outside Brazil. > > Any thoughts, anyone?
From: G. A. Edgar on 13 Sep 2009 13:14 In article <59c30936-78bb-4dd4-8801-f9c6c2ea09e5(a)z24g2000yqb.googlegroups.com>, Br�ulio Bezerra <brauliobezerra(a)gmail.com> wrote: > Hi, > > Virtually every kid in Brazil learns to divide polynomials by one of > the form (x - a) using the Briot-Ruffini method. It is just a long > division method, but works a little faster since someone writes only > the coefficients. It also has numerical applications, since it can be > used to find the value of a polynomial and its derivatives of all > orders at some given point, in a particularly efficient way. This is > done viewing the polynomial as (((a_4x + a_3) x + a_2) x + a_1 ) x + > a_0, for example. > > The problem is that this method is only mentioned by Brazilian people > in books, on the Internet, everywhere. So I don't know if this method > is so obvious that it commonly deserves no mention; if this is just a > trend we have in Brazilian education; or if it has another name > outside Brazil. I would say: "synthetic division" http://mathworld.wolfram.com/SyntheticDivision.html -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/
From: Chip Eastham on 13 Sep 2009 13:20 On Sep 13, 1:14 pm, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid> wrote: > In article > <59c30936-78bb-4dd4-8801-f9c6c2ea0...(a)z24g2000yqb.googlegroups.com>, > > > > Bráulio Bezerra <brauliobeze...(a)gmail.com> wrote: > > Hi, > > > Virtually every kid in Brazil learns to divide polynomials by one of > > the form (x - a) using the Briot-Ruffini method. It is just a long > > division method, but works a little faster since someone writes only > > the coefficients. It also has numerical applications, since it can be > > used to find the value of a polynomial and its derivatives of all > > orders at some given point, in a particularly efficient way. This is > > done viewing the polynomial as (((a_4x + a_3) x + a_2) x + a_1 ) x + > > a_0, for example. > > > The problem is that this method is only mentioned by Brazilian people > > in books, on the Internet, everywhere. So I don't know if this method > > is so obvious that it commonly deserves no mention; if this is just a > > trend we have in Brazilian education; or if it has another name > > outside Brazil. > > I would say: "synthetic division" > > http://mathworld.wolfram.com/SyntheticDivision.html That's the term I'd use. Note that Horner's method for evaluating polynomial p(X) at X = a uses the same intermediate computations as synthetic division of p(X) by X-a, but retains only the "remainder" r: p(X) = (X-a)q(X) + r regards, chip
From: Bill Dubuque on 13 Sep 2009 13:38
Br�ulio Bezerra <brauliobezerra(a)gmail.com> wrote: > > Virtually every kid in Brazil learns to divide polynomials by one of > the form (x - a) using the Briot-Ruffini method. It is just a long > division method, but works a little faster since someone writes only > the coefficients. It also has numerical applications, since it can be > used to find the value of a polynomial and its derivatives of all > orders at some given point, in a particularly efficient way. This is > done viewing the polynomial as (((a_4x + a_3) x + a_2) x + a_1 ) x + > a_0, for example. > > The problem is that this method is only mentioned by Brazilian people > in books, on the Internet, everywhere. So I don't know if this method > is so obvious that it commonly deserves no mention; if this is just a > trend we have in Brazilian education; or if it has another name > outside Brazil. > > Any thoughts, anyone? Google "synthetic division". Like many manual computational techniques (e.g. sqrt by hand) it's gone out of fashion in the modern computer era. But a Google Books search shows that it still occurs in various places. Perhaps you might enjoy one of my old observations: Note that the Horner poly form is also useful elsewhere, e.g. powering by repeated squaring arises from rewriting the expt as a Horner poly in radix 2, e.g. as below, to compute x^101 1 1 0 0 1 0 1 = 101 in binary notation = 1)2+1)2+0)2+0)2+1)2+0)2+1 = 101 as binary Horner poly 2 2 2 2 2 2 101 -> x) x) ) ) x) ) x = x via repeated squaring So here we see examples where it is advantageous to not apply the distributive law to fully expand a poly to normal form. Do you know other such examples? --Bill Dubuque |