Palindromic polynomial

Palindromic polynomial

A polynomial is palindromic, if the sequence of its coefficients are a palindrome.

Let P(x) = sum_{i=0}^n a_ix^i be a polynomial of degree n, then P is palindromic if a_i = a_{n-i} for i=0...n.

Similarly, P is called antipalindromic if a_i = -a_{n-i} for i=0...n.


Some examples of palindromic polynomials are:

(x+1)^2 = x^2 + 2x + 1

(x+1)^3 = x^3 + 3x^2 + 3x + 1

Generally, the expansion of (x+1)^n is palindromic for all n (can see this from binomial expansion)

It also follows that if P is of even degree (so has odd number of terms in the polynomial), then it can only be antipalindromic when the 'middle' term is 0, i.e. a_i=-a_i, where n=2i.

ee also

* Reciprocal polynomial

External links

* [ MathPages - The Fundamental Theorem for Palindromic Polynomials]

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Reciprocal polynomial — In mathematics, for a polynomial p with complex coefficients,:p(z) = a 0 + a 1z + a 2z^2 + ldots + a nz^n ,!we define the reciprocal polynomial, p*:p^*(z) = overline{a} n + overline{a} {n 1}z + ldots + overline{a} 0z^n = z^noverline{p(ar{z}^{… …   Wikipedia

  • Jones polynomial — In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1983. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Palindrome — Palindromes redirects here. For the film, see Palindromes (film). See also: Constrained writing A palindrome is a word, phrase, number, or other sequence of units that can be read the same way in either direction, with general allowances for… …   Wikipedia

  • Палиндром — Запрос «Перевёртыш» перенаправляется сюда; см. также другие значения. Палиндром (от греч. πάλιν  «назад, снова» и греч. δρóμος  «бег»), иногда также палиндромон, от гр. palindromos бегущий обратно[1])  число (например, 404),… …   Википедия

  • Line spectral pairs — (LSP) or Line Spectral Frequencies (LSF) are used to represent Linear Prediction Coefficients (LPC) for transmission over a channel. LSPs have several properties (e.g. smaller sensitivity to quantisation noise) that make them superior to direct… …   Wikipedia

  • Polynôme réciproque — En mathématiques, le polynôme réciproque d un polynôme à coefficients complexes est le polynôme P* défini par : où désigne le conjugué de a. Pour tout nombre complexe z non nul, on a donc  …   Wikipédia en Français

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • 500 (number) — For other uses, see 500 (disambiguation). ← 499 501 → 500 List of numbers Integers …   Wikipedia

  • Возвратное уравнение — Алгебраическое уравнение вида: называется возвратным, если его коэффициенты, стоящие на симметричных относительно середины позициях, равны, то есть если , при k = 0, 1, …, n. Содержание 1 Уравнение четвёртой степени …   Википедия

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”