>On Jul 30, 2005, at 10:02 AM, Kirby Urner wrote: > >> We're talking about phi ('fie' -- not 'pi' and not 'fee'). > >Actually, a scholar in classics once told me that it's 'fee'--and ought to >be 'pee' rather than 'pie'.
Well, if we are going to split hairs, it's "phee", but who cares!
Back to the polynomial, there are two contexts where it arises. First is the classical derivation of the Golden Mean, as Kirby points out. But there is another pseudo-classical use. The two roots (actually, halves of the two roots, if I recall correctly) generate the Fibonacci sequence.
I'll leave you with this point as an exercise.
But back to Kirby's question, my answer would simple. We could just as well take another polynomial to establish some sort of a special or classical relationship. The question really should be, "How is the application of this polynomial important to the student's learning progress?" The issue should not be of the importance of the particular piece in the *history* of mathematics, but its importance in the learning process. There are no irreplaceable parts in the curriculum, but many of the ideas we want students to learn may well be irreplaceable. The trouble is, we cannot agree on which ideas these are.
Kirby, if you want to use a particular polynomial as the cornerstone of some part of the curriculum, more power to you. But it does not mean that everyone (or anyone) should follow your example.