EDB — 09J

Exercises

1. [09J]Let \(K\) be a positive integer, \(N\) an integer, and \(I=\{ N,N+1,\ldots ,N+K\} \) be the sequence of integers from \(N\) to \(N+K\). For each \(n∈ I\) we set an integer values \(a_ n\). Let \(p\) be the only one polynomial of degree \(K\) such that \(p(n)=a_ n\) for every \(n∈ I\).

   • Show that \(p\) has rational coefficients.

   • Show that \(p(x)\) is integer for every \(x\) integer.

   • Find an example of a polynomial \(p\) which takes integer values for \(x\) integer, but not all coefficients of \(p\) are integers.

   • What happens if \(I\) contains \(K+1\) integers, but not consecutive? Is it still true that, defining \(p(x)\) as above, \(p\) only assumes integer values on integers?