Let P(x) be a polynomial such that
(x+1)P(x-1)=(x-1)P(x)
for all real values of x. Determine the maximum possible degree of P(x).
Start by substituting x=-1 in the identity and simplify. Then P(-1) = 0. Also substitute x=1 in the identity and simplify. Then P(0) = 0. Then x and x+1 are factors of P(x).
Let Q(x) be a polynomial such that P(x) = (x+1)*x*Q(x). Substitute this into the identity. Then (x+1)*x*(x-1)*Q(x-1) = (x-1)*(x+1)*x*Q(x), which reduces to Q(x-1)=Q(x) for any x other than -1, 0, or 1.
The only polynomial Q(x) that satisfies this condition is the constant polynomial. Then P(x) = k*(x+1)*x for some constant k, which means that P(x) has a maximum possible degree of two.
Solution
