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# 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, 1, ...

*n*.

Similarly, P is called **antipalindromic** if *a _{i}* = −

*a*

_{n − i}for

*i*= 0, 1, ...

*n*. It follows from the definition 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*, where

_{n}*n*= 2

*i*.

Some examples of palindromic polynomials are:

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

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

These are examples of the expansion of \( (x+1)^n \), which is palindromic for all *n*, this can be seen from the binomial expansion.

Another example of a palindromic polynomial [which isn't of the form \( (x+1)^n \)] is:

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

An example of an antipalindromic polynomial is:

\( x^2 - 1 \)

Note the zero coefficient for the term in x.

Properties

- If
*a*is a root of a polynomial that is either palindromic or antipalindromic, then 1/*a*is also a root and has the same multiplicity.^{[1]} - The converse is true: If a polynomial is such that if
*a*is a root then 1/*a*is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic. - The product of two palindromic or antipalindromic polynomials is palindromic.
- The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
- A palindromic polynomial of odd degree is a multiple of
*x*+1 (it has -1 as a root) and its quotient by*x*+1 is also palindromic. - An antipalindromic polynomial is a multiple of
*x*-1 (it has 1 as a root) and its quotient by*x*-1 is palindromic. - An antipalindromic polynomial of even degree is a multiple of
*x*^{2}-1 (it has -1 and 1 as a roots) and its quotient by*x*^{2}-1 is palindromic. - If
*p*(*x*) is a palindromic polynomial of even degree 2*d*, then there is a polynomial*q*of degree*d*such that*x*^{d}*q*(*x*+1/*x*) =*p*(*x*).

It results from these properties that the study of the roots of a polynomial of degree *d* that is either palindromic or antipalindromic may be reduced to the study of the roots of a polynomial of degree at most *d*/2.

Factorization

Factorization techniques (and the search for roots) follow on directly from the properties listed above.

For example, Property 5 yields an immediate factor *x*+1 for palindromic polynomials of odd degree.

As another example, Property 8 leads to the technique of dividing by *x*^{d} and replacing *x* + *1*/*x* by *X*.

As an example of the latter technique suppose

\( x^4 + x^2 + 1 = 0 \)

Letting X = x + 1/x, dividing by \( x^2 \) and deriving

\( X^2 = x^2 + 2 + 1/x^2 \)

we have the much simpler

\( X^2 - 1 = 0 \)

which factorizes as

\( (X - 1)(X + 1) = 0 \)

so either X = 1 or X = - 1

The X = - 1 case yields

x + 1/x = - 1

or

\( x^2 + x + 1 = 0 \)

which has no real roots.

The X = 1 case yields

\( x + 1/x = 1 \)

or

\( x^2 - x + 1 = 0 \)

which also has no real roots.

Converting other polynomials to palindromic form

Some polynomials can be converted to palindromic form by, for example, suitable substutions. For example consider

\( 4x^2 + 4x + 1. \)

Writing y = 2x this becomes

\( y^2 + 2y + 1 \)

or

\( (y + 1)^2 \)

with the resultant factorization

\( (2x + 1)^2 \)

Similar techniques might yield a polynomial in antipalindromic form.

See also

Reciprocal polynomial

Notes

Pless 1990, pg. 57 for the palindromic case only

External links

"The Fundamental Theorem for Palindromic Polynomials" at MathPages.com.

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