Consider a sequence of polynomials with real coefficients defined by

Question

Consider a sequence of polynomials with real coefficients defined by:

$$p_0=(x^2 +1)(x^2 +2).....(x^2 +1009)$$

with subsequent polynomials defined by $$p_{k+1} (x) :=p_k (x+1) - p_k (x) $$ for $x>0$. Find the least n such that

$$p_n (1)=p_n (2)=......=p_n (5000).$$

My attempt :

Degree of the first given polynomial is 2018. And it can be seen that for $p_n (x) =2018-n$. So to have 5000 roots it must be constant function. So n=2018.

Answer 1

Combine the following:

• $p_0(x)$ has degree $2018$.
• $\deg p_{k+1}=\deg p_k-1$ for all $k$.
• For $p_n(x)-p_n(1)$ to have $5000$ zeros it must have either degree $\ge5000$ or be the constant zero.