Determine all polynomials L(x) satisfying the undernoted equation:
(x-243) L(3x) = 243 (x-1) L(x) for all x.

L(x) is a polynomial with (x-243)L(3x)=243(x-1)L(x) for all x.
Substituting x=243 and x=1 gives L(243)=0 and L(3)=0. Thus, L(x) can be written as
(x-3)(x-243)P(x) where P(x) is a polynomial.

Substituting this into the original identity gives
(x-81)P(3x)=27(x-3)P(x).
Substituting x=81 and x=3 gives P(81)=0 and P(9)=0. Thus, L(x) can be written as
(x-3)(x-243)(x-81)(x-9)Q(x) where Q(x) is a polynomial.

Substituting this into the original identity gives
(x-27) Q(3 x)=3(x-9) Q(x).
Substituting x=27 and x=9 gives Q(27)=0 and Q(27)=0. Thus, L(x) can be written as
(x-3)(x-243)(x-81)(x-9)(x-27)R(x) where R(x) is a polynomial.

Substituting this into the original identity gives
R(3 x)=R(x).
The only polynomials that meets this identity are the constant polynomials.
Thus, L(x)=c(x-3)(x-9)(x-27)(x-81)(x-243), where c is any complex number.

Edited on December 23, 2005, 10:22 pm

Posted by owl on 2005-12-23 22:16:40