# Palindromic polynomial

Palindromic polynomial

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

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

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

Examples

Some examples of palindromic polynomials are:

$\left(x+1\right)^2 = x^2 + 2x + 1$

$\left(x+1\right)^3 = x^3 + 3x^2 + 3x + 1$

Generally, the expansion of $\left(x+1\right)^n$ 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

* [http://www.mathpages.com/home/kmath294.htm MathPages - The Fundamental Theorem for Palindromic Polynomials]

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