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 \( m \).


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us