be a polynomial of degree n, then P is palindromic if ai = an − i for i = 0, 1, ... n.
Similarly, P is called antipalindromic if ai = −an − 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. ai = −an, where n = 2i.
Some examples of palindromic polynomials are:
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:
An example of an antipalindromic polynomial is:
Note the zero coefficient for the term in x.
- 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.
- 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 x2-1 (it has -1 and 1 as a roots) and its quotient by x2-1 is palindromic.
- If p(x) is a palindromic polynomial of even degree 2d, then there is a polynomial q of degree d such that xdq(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 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 xd and replacing x + 1/x by X.
As an example of the latter technique suppose
Letting X = x + 1/x, dividing by x2 and deriving
we have the much simpler
which factorizes as
so either X = 1 or X = - 1
The X = - 1 case yields
which has no real roots.
The X = 1 case yields
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
Writing y = 2x this becomes
with the resultant factorization
Similar techniques might yield a polynomial in antipalindromic form.
- Pless 1990, pg. 57 for the palindromic case only
- "The Fundamental Theorem for Palindromic Polynomials" at MathPages.com.