  Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge
2$, it is conjectured that the number of preperiodic points of $f$ is bounded
by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the
only examples of parametric families of polynomials with no preperiodic points
are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In this
article, given any integer $d\ge 2$, we display infinitely many parametric
families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no
rational preperiodic points for any $t\in K$.
