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_n, where n = 2i.

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

1. 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]
2. 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.
3. The product of two palindromic or antipalindromic polynomials is palindromic.
4. The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
5. 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.
6. An antipalindromic polynomial is a multiple of x-1 (it has 1 as a root) and its quotient by x-1 is palindromic.
7. An antipalindromic polynomial of even degree is a multiple of x²-1 (it has -1 and 1 as a roots) and its quotient by x²-1 is palindromic.
8. If p(x) is a palindromic polynomial of even degree 2d, then there is a polynomial q of degree d such that x^dq(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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