algebra precalculus - Integer coefficients polynomial. Find largest number of roots.

The polynomial $p(x)$ has integer coefficients, and $p(100)=100$. Let $r_1, r_2, …, r_k$ be distinct integers that satisfy the equation $p(x)=x^3$. What is the largest possible value of $k$?

Answer

Suppose there are $k$ distinct integer roots for $p(x)-x^3$. Then we may write $p(x) = x^3+q(x)\prod_{i=1}^k(x-r_i) \implies 100 = 100^3+q(100)\prod_{i=1}^k(100-r_i)$.

This gives $q(100) \prod_{i=1}^k(100-r_i) = -999,900=-2^2\cdot3^2\cdot5^2\cdot 11 \cdot 101$

LHS is thus a product of $k+1$ integers of which at least $k$ are distinct, and the RHS can be expressed as a product of at most $11$ factors. Hence $k \le 10$.

To prove $k_{max} = 10$, all we need now is to demonstrate one polynomial $p(x)$, say:
$$100^3-(x-99)(x-101) (x-102) (x-98) (x-103) (x-97) (x-105) (x-95) (x-111) (x-201)$$
which will satisfy the conditions.