## Polynomials with Rational Roots that Differ by a Non-zero Constant

**Authors:** Philip Gibbs

The problem of finding two polynomials P(x) and Q(x) of a given degree n in a single variable
x that have all rational roots and differ by a non-zero constant is investigated. It is shown
that the problem reduces to considering only polynomials with integer roots. The cases n < 4
are solved generically. For n = 4 the case of polynomials whose roots come in pairs (a,-a) is
solved. For n = 5 an infinite number of inequivalent solutions are found with the ansatz
P(x) = -Q(-x). For n = 6 an infinite number of solutions are also found. Finally for n = 8
we find solitary examples. This also solves the problem of finding two polynomials of degree
n that fully factorise into linear factors with integer coefficients such that the difference
is one.

**Comments:** 6 pages

**Download:** **PDF**

### Submission history

[v1] 24 Aug 2009

[v2] 25 Aug 2009

**Unique-IP document downloads:** 408 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary.
In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution.
Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*