Number Theory


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

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Submission history

[v1] 24 Aug 2009
[v2] 25 Aug 2009

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