
Which Polynomial?
Posted:
Jul 30, 2005 12:02 PM


We'd likely consider it ludicrous were NCLB administrators to put out that they had a special polynomial in mind, for all school children of eligible age to solve beginning next school year. AntiNCLBers would scoff that you don't "solve" a polynomial because polynomials needn't equate to anything and therefore the matter of "solving for an unknown" (perhaps several) is moot. Others would simply decry the dictatorial sound of "mandating a polynomial."
However, *were we* to accept the above surreal development (the NCLB's mandating of an Official Polynomial), it'd be interesting to speculate just *which* polynomial (equated to zero) that'd be. I have a candidate:
p**2  p  1 = 0.
Why?
First, let's look at a solution. Either completing the square or the quadratic formula suggest themselves. Solve it both ways. Your answers will be (1+pow(5,0.5))/2 and (1pow(5,0.5))/2 [Python notation]. Only one of those is positive, so if looking for a geometric length, the positive root is the most relevant. And that solves as a decimal to around 1.618, which some of you may recognize. We're talking about phi ('fie'  not 'pi' and not 'fee').
So where does this polynomial come from? From a diagram:
p q 
The goal here is to establish this relationship:
(q:p) :: (p:(p+q))
or
q/p == p/(p+q)
Set q = 1 and solve for p.
phi is important as the ratio between a side and a diagonal of a regular pentagon. It is known as the "golden mean" and has been called that since ancient times. That should be your cue: this number *is* important. That's why the NCLB is on board. Don't leave any child behind next year. How 'bout it folks?
Kirby

