Given that x is a real number, determine all polynomials with real coefficients satisfying this relationship:
P(x2) = P(x)*P(x+2)
Among constant functions P(x)=0
and P(x)=1 are clearly the only possibilities.
Among linear functions P(x)=ax+b
a little algebra shows a=1, b=-1 is the only possibility.
Among quadratics P(x)=ax²+bx+c
a lot of algebra shows a=1, b=-2, c=1 is the only possibility.
From which appears a nice generalization:P(x)=(x-1)^n for all non-negative integers n.
(x^2 -1)^n = (x-1)^n*(x+1)^n
above are the solutions.
I did not check for other higher order polynomials. It seems these are the only ways to make the zeros to work out right.
Posted by Jer
on 2015-06-27 16:40:37