## Puzzle: a problem with polynomials

The Open University equivalent of DoF is the M500 magazine, the publication of the M500 Society , a self-help group, which “helps to mitigate student isolation by means of the M500 Magazine and the Mathematics Weekends”. It is not clear to me one can subscribe to it not being a student in the OU, but their membership page is worth visiting, it is a hoot.

The magazine has a variety of problems, some of them easy, some of them pretty hard. I will be delving into this treasury from time to time…

Problem 242.2: show that the real root of $x^3-x-1=0$ is also a root of $x^5-x^4-1=0$ without solving these equations.

(Of course you could find the real root of the cubic by Tartaglia’s method, substitute it into the quintic, sweat a bit and so on; but there is no need to sweat.) The reason that I am suggesting this elementary problem is that while discussing its solution, I will be able to introduce something that is not that elementary and should be better known, viz., resultants.

(MG)