Show that RP is closed under concatenation
Do the following $poly(|x|)$ times:


  Check for every possible (contiguous) partition of the input string $x = x_1x_2$ and apply the $\mathrm{RP}$ machine to each of $x_i$ if the answer is $\mathrm{YES}\:\mathrm{YES}$ outputs $\mathrm{YES}$ and halt.


If after polynomially many times, still no definite $\mathrm{YES}$ answer has been put down, output $\mathrm{NO}$ with a little bit of fear as it is always like so with $\mathrm{RP}$ algorithms.