We show that the prime divisors of a random polynomial in F-q[t] are typically "Poisson distributed". This result is analogous to the result in Z of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdos-Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {omega(f)}(f is an element of Fq)[t] than was previously known, where omega(f) is the number of prime divisors of f.