1. Let where is an integer. Prove that cannot be expressed as the product of two polynomials, each of which has all its coefficients integers and degree at least 1.
2. Let be a polynomial with integer coefficients satisfying . Show that has no integer zeros.
3. Let be a polynomial with real coefficients. Show that there exists a nonzero polynomial with real coefficients such that has terms that are all of a degree divisible by .