Polynomials with Integer Coefficients

Consider a polynomial \(P(x)=a_nx^n+\cdots+a_1x+a_0\) with integer coefficients. The difference \(P(x)-P(y)\) can be written in the form \[a_n(x^n-y^n)+\cdots+a_2(x^2-y^2)+a_1(x-y),\] in which all summands are multiples of polynomial \(x-y\). This leads to the simple though important arithmetic property of polynomials from \(\mathbb{Z}[x]\):

Theorem 3.1

If \(P\) is a polynomial with integer coefficients, then \(P(a)-P(b)\) is divisible by \(a-b\) for any distinct integers \(a\) and \(b\).

In particular, all integer roots of \(P\) divide \(P(0)\).

There is a similar statement about rational roots of polynomial \(P(x)\in\mathbb{Z}[x]\).

Theorem 3.2

If a rational number \(p/q\) (\(p,q\in\mathbb{Z}\), \(q\neq 0\), gcd\((p,q)=1\)) is a root of polynomial \(P(x)=a_nx^n+\cdots+a_0\) with integer coefficients, then \(p\mid a_0\) and \(q\mid a_n\).

Problem 6

Polynomial \(P(x)\in\mathbb{Z}[x]\) takes values \(\pm1\) at three different integer points. Prove that it has no integer zeros.

Problem 7

Let \(P(x)\) be a polynomial with integer coefficients. Prove that if \(P(P(\cdots P(x)\cdots))=x\) for some integer \(x\) (where \(P\) is iterated \(n\) times), then \(P(P(x))=x\).

Note that a polynomial that takes integer values at all integer points does not necessarily have integer coefficients, as seen on the polynomial \(\frac{x(x-1)}2\).

Theorem 3.3

If the value of the polynomial \(P(x)\) is integral for every integer \(x\), then there exist integers \(c_0,\dots,c_n\) such that \[P(x)=c_n\binom xn+c_{n-1}\binom x{n-1}+\cdots+c_0\binom x0.\] The converse is true, also.

Problem 8

Suppose that a natural number \(m\) and a real polynomial \(R(x)=a_nx^n+ a_{n-1}x^{n-1}+\dots+a_0\) are such that \(R(x)\) is an integer divisible by \(m\) whenever \(x\) is an integer. Prove that \(n!a_n\) is divisible by \(m\).