- #1
tdenise
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Homework Statement
Find all the polynomials [itex]P(x)[/itex] for which
[tex]P(x^2+2x+3)=[P(x+3)]^2[/tex]
Homework Equations
The Attempt at a Solution
I don't really know how to solve functional equations systematically. I tried to to find a linear [itex]P(x)[/itex] and found [itex]P(x)=x-2[/itex] through trial & error. I also tried substituting x = 0 and x = (x-3) but that didn't go anywhere.
[tex]x=0: P(3) = [P(3)]^2[/tex]
[tex]\Rightarrow{P(3) = {0,1}}[/tex]
[tex]x=(x-3): P((x-2)^2+2) = [P(x)]^2[/tex]
Note that on the LHS, there is [itex](x-2)[/itex] (the P(x) I found) but I don't know if that's a coincidence or not. Also the LHS & RHS are ≥ 0 and [itex]P(x)[/itex] must be an "odd degree" polynomial (i.e. the leading term must be an odd power).
Please help me solve this question. Thanks.