Whitehead's point-free geometry

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

Let

$P(x)=\sum _{i=0}^{n}a_{i}x^{i}$

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.

Examples

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.
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 x² 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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