ISI2025-MCS-PCB (Non-CS) | Question: 8

Question

Let $p(x)=(x-1)(x-2) \cdots(x-n+1)(x-n)-1$ be a polynomial where $n \geq 1$. Suppose $p(x)=f(x) g(x)$ where $f$ and $g$ are both polynomials with integer coefficients.

1. Show that $f(k)=-g(k)$ for all $k=1,2, \ldots, n$.
2. Show that either $f(x)=-g(x)$ for all $x \in \mathbb{R}$, or one of $f$ and $g$ is a constant polynomial.
3. Subsequently, prove that $p(x)$ cannot be written as a product of two non-constant polynomials with integer coefficients.