Re: Mathematica: Long divison for polynomials

*To*: mathgroup at smc.vnet.net*Subject*: [mg83816] Re: [mg83772] Mathematica: Long divison for polynomials*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 1 Dec 2007 05:45:33 -0500 (EST)*References*: <200711301018.FAA04867@smc.vnet.net>

On 30 Nov 2007, at 19:18, Caren Balea wrote: > Dear all, > > I would like to perform long division for a polynomial > using Mathematica. > I came across the command "PolynomialQuotient". > > If I consider a function > f(x) = g(x)/h(x) > where degree g(x) > h(x) then I can use PolynomialQuotient > to do the job - no problem. > > Now, say, I have > f(x) = 1/h(x) > where > h(x) = 1 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 > and I want to do long division for f(x). > > So, I enter > > g[x_] := 1 > h[x_] := 1 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 > k[x_] = PolynomialQuotient[g[x], h[x], x] > > and obtain > 0. > > Which is correct - *but* > g(x)/h(x) can be also written as > g(x)/h(x) = 1 - a1*x + (a1^2 - a2)*x^2 + ... > > How do I obtain the last expression? > Which command do I need to use? > > Thank you, > Caren > Normal[1/(1 + a1*x + a2*x^2 + a3*x^3 + a4*x^4) + O[x]^4] (-a1^3 + 2*a2*a1 - a3)*x^3 + (a1^2 - a2)*x^2 - a1*x + 1 If you want the general term you can try (in Mathematica 6) SeriesCoefficient[1/(a4 x^4 + a3 x^3 + a2 x^2 + a1 x + 1), {x, 0, n}] though I have no idea if the complicated answer returned is correct. Andrzej Kozlowski