Write a function that returns the coefficient of x^n

Question

Write a function C[p_, x_, n_] that returns the coefficient of $x^n$ in the polynomial equation.

C[p_, x_, n_] := ...

If we call C[7 x^2 - 3 x^3, x, 2], the output should be 7. I don't want to use the Coefficient[] function in Mathematica, I just want to understand how it is done.

Is this a programming exercise from school, or something you're just wondering about? — J. M.'s missing motivation, Oct 29 '15 at 06:50
Nothing, actually, I'm practising by solving exercises but on this one, i'm just completely stuck. From school, it's not a homework, it is in our book — ferrou, Oct 29 '15 at 06:51
Perhaps start by taking a look at the FullForm of a polynomial in Mathematica. You could then try your hand at some pattern matching using Cases... — MarcoB, Oct 29 '15 at 06:52
Is it acceptable to differentiate $n-1$ times and then evaluate the polynomial at $0$? — Patrick Stevens, Oct 29 '15 at 07:12

score 6 · Accepted Answer · answered Oct 29 '15 at 07:40

6

Just do what SeriesCoefficient does yourself

coeff[polynomial_, variable_, order_] := 
 D[polynomial, {variable, order}]/(order!) /. variable -> 0

coeff[7 x^2 - 3 x^3, x, 2]
(* 7 *)

coeff[347 x^19 - 7 x^2 - 3 x^3, x, 19]
(* 347 *)

answered Oct 29 '15 at 07:40

Jason B.

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If one wanted to be cruel, he could have posted a contour integral method for extracting the coefficient… and then there's the method that uses FFT. – J. M.'s missing motivation Oct 29 '15 at 07:44
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There should be a badge or something for the most convoluted answer to a question. Or the most inscrutable - some of the answers here are so compact I have no clue what they are doing – Jason B. Oct 29 '15 at 07:48
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@JasonB I suppose one could propose it as a Mathematica-specific popularity contest on [codegolf.SE] – David Z Oct 29 '15 at 11:37
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How about a prize for answers as short as a Tweet? – David G. Stork Oct 29 '15 at 16:54

march · Answer 2 · 2016-01-13T16:34:31.630

2

Here's a pattern-matching version (not tested carefully since the question is already answered):

coeff[poly_, var_, orders_List] := Cases[Expand@poly, a_. var^# :> a] & /@ orders

Usage: orders is a list of the wanted orders, so

poly = (x + 5) (x - 1) (x^2 + y); Expand@poly
coeff[poly, x, {3, 1, 8, 4}]
(* -5 x^2 + 4 x^3 + x^4 - 5 y + 4 x y + x^2 y *)
(* {{4}, {4 y}, {}, {1}} *)

edited Jan 13 '16 at 16:34

answered Oct 29 '15 at 16:22

march

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Write a function that returns the coefficient of x^n

2 Answers2