Ideal multiplication in polynomial rings

Question

Let $ K = \mathbb{C}[x_1, \dots, x_n] $ where $ n > 1 $. Say I have two non-principal ideals $ I $ and $ J $, where $ I = (f_1, \dots, f_r) $ and $ J = (g_1, \dots, g_s) $. Then I know that $ IJ = (\prod f_i g_j) $, but is it ever possible for $ IJ $ to be principal? I'm also interested in the specific case where $ I $ is simply $ (x_1, \dots, x_n ) $.