Find all pairs of polynomials P(x),Q(x) with integer coefficients such that P(Q(x)) = (x - 1)(x - 2)...(x - 9) for all real numbers x
Call G(x)=
(x - 1)(x - 2)...(x - 9)
The simplest solutions are Q(x)=G(x), P(x)=x
or Q(x)=x, P(x)=G(x)
But then you can make P(x)=G(x-n), Q(x)=x+n for any integer n.
This is just a shift.
You could use any invertible function that covers the reals. But to keep then polynomials with integer coefficients is more limiting.
I'd love to see both P and Q be third degree polynomials, but nothing I've tried seems to work.
|
|
Posted by Jer
on 2021-05-20 14:50:14 |
Some simple pairs