Synthetic, or man-made, elements typically exist for only a short time

This is true. Elements past lead are radioactive, because the repulsive force of the protons cannot be overpowered by the "gluing" ability of neutrons (remember, likes repel). As more and more protons are added, generally, the elements become more unstable; for example, Bismuth, right next to lead on the Periodic Table, is radioactive, but the half life of this element is about a billion times longer than the current age of the universe, but Oganesson, element number 118, has a half life of fractions of a second.