Let P(x) be a nonzero polynomial such that (x-1)P(x+1) = (x+2)P(x) for every real x, and P(2)^2 = P(3). Then find P(2016).
Set x=1. Then 0 = 2*P(1), so x-1 is a factor of P(x)
Set x=-2. Then -3*P(-1) = 0, so x+1 is a factor of P(x)
Set x=0. Then -1*P(1) = 2*P(0), and we know P(1)=0. Then 0 = 2*P(0), so x is a factor of P(x).
At this point let P(x) = Q(x)*(x-1)*x*(x+1). Then the identity becomes
(x-1)*Q(x+1)*x*(x+1)*(x+2) = (x+2)*Q(x)*(x-1)*x*(x+1)
This reduces to Q(x) = Q(x+1). The only polynomial which satisfies this condition is the constant polynomial.
Then P(x) = C*(x-1)*x*(x+1) for some constant C.
Now the extra constraint comes into play
P(2) = C*1*2*3 = 6C and P(3) = C*2*3*4 = 24C; then P(2)^2 = P(3) becomes (6C)^2 = 24C. The nonzero solution is C=2/3.
P(x) = (2/3)*(x-1)*x*(x+1).
Then P(2016) = (2/3)*2015*2016*2017 = 5462358720.
Analytic Solution
