Let $(b_n) = (b_1, b_2, ...)$ be a sequence of integers. A primitive prime
divisor of a term $b_k$ is a prime which divides $b_k$ but does not divide any
of the previous terms of the sequence. A zero orbit of a polynomial $f(z)$ is a
sequence of integers $(c_n)$ where the $n$-th term is the $n$-th iterate of $f$
at 0. We consider primitive prime divisors of zero orbits of polynomials. In
this note, we show that for integers $c$ and $d$, where $d > 1$ and $c \neq \pm
1$, every iterate in the zero orbit of $f(z) = z^d + c$ contains a primitive
prime whenever zero has an infinite orbit. If $c = \pm 1$, then every iterate
after the first contains a primitive prime.