Let P(x) be a polynomial of degree 2m.
Show that $ P(x^2)= P(x)P(x-1) $ if and only if
$$ P(x) =\left( x^2 + x+1 \right)^m$$

Question

Let P(x) be a polynomial of degree 2m.
Show that $ P(x^2)= P(x)P(x-1) $ if and only if
$$ P(x) =\left( x^2 + x+1    \right)^m$$

anonymous · Answer

See http://openstudy.com/study#/updates/4ff6b092e4b01c7be8c98dee for the particular case where m=5.

anonymous · Answer

hmm looks like all the heavy lifting was done showing that since the only roots are $-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$ it must be $(x^2+x+1)^m$ 
is that the hook, or is there something else?

anonymous · Answer

I agree. I posted so people can see that it is true in this general form.

anonymous · Answer

ok then kudos to @mukushla  who did all the work for this one