Express $f(n)$ in terms of $f(1),f(0)$, and conclude by infinite roots. Lazy to do it though sorry

jasperE3 wrote: Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$ We have that $$2P(x)+2=P(x+1)+P(x-1) \iff P(x)-P(x-1)-2x=P(x+1)-P(x)-2(x+1)$$ Hence, the polynomial $P(x)-P(x-1)-2x$ is constant over the integers. so it must be constant over the reals. Let $$P(x)-P(x-1)-2x=c \quad (i)$$ . Then for all integers $n$, we have by summing up $(i)$ for $x=1,2, \cdots , n$ that $$P(n)=n(n+c+1)+P(0)$$ . The above equation has infinite roots, so it must be true for all reals as well. hence $P$ is a quadratic polynomial. on checking we find that all of them work.

https://youtu.be/iytNs9SCBJs?t=608