# 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 $$p/q$$ ($$p,q\in\mathbb{Z}$$, $$q\neq 0$$, nzd$$(p,q)=1$$) 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 $$m$$.

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