148 4. Equations- Let f(t) be a polynomial of degree n over Z. Show that, if f(t) as-
sumes prime integer values for 2n + 1 distinct values of t, then f(t)
is irreducible over Z.
Explorations
E.41. Zeros of the Derivative. Suppose a polynomial f(t) of degree n
over R has n real zeros. Since every zero of f(t) of multiplicity m is a
zero of f’(t) of multiplicity m - 1, and since, by Rolle’s Theorem, there
is a zero of f’(t) b e t ween any distinct pair of consecutive zeros of f(t), it
follows that f’(t) has at least n - 1 real zeros counting multiplicity. Since
degf’(t) = n - 1, it follows that all the zeros of f’(t) are real.
The fact that the reality of all zeros of a polynomial implies the reality
of all zeros of its derivative can be formulated in a way which leads to
an interesting generalization. If u is the smallest and v the largest zero of
f(t), then not only the zeros of f(t) but also those of f’(t) lie in the closed
interval [u, w].
Now suppose that f(t) is an arbitrary polynomial over C of degree n with
zeros r1,r2,..., z n (not necessarily all distinct). These are represented by
points in the complex plane. Let P be the smallest convex polygonal region
with boundary which contains them; some of the zeros will be vertices,
others may lie on edges while the remainder will be in the interior. The
diagram illustrates a possible situation:
Observe that this polygonal region can be represented as the intersection
of half-planes, namely those portions of the complex plane which lie on one
side of the lines containing the edges of the polygon.