# Polynomial with random roots

Hypothesis: all polynomials and operations are defined over a finite field, $\mathbb{F}_p$, where $p$ is a prime number.

In the literature, especially in cryptography, a random polynomial defined as a polynomial whose coefficients are distributed uniformly at random over a field, e.g. $\mathbb{F}_p$. A simple way of constructing a random polynomial of degree $d$ is to pick $d+1$ random values from the field and build a polynomial.

Question: Is $f=\prod^{d}_{i=1}(x-r_i)$ a random polynomial, where $r_i$'s are picked uniformly random fron the field?

For example only about half of the polynomials of the form $x^2+c$ are products of linear factors.
And of course by multiplying together $x-r_i$ factors you can only ever get monic polynomials.
In general, observe that both the set of ordered sequences of $d$ roots and the set of monic polynomials of degree $d$ have the same cardinality, but the mapping from roots to polynomials is not injective since the ordering of the root doesn't matter. So some of the polynomials must be missed by the mapping when $d>1$.