The chemical elements labelled as "synthetic" are unstable, with a half-life
so short (from a few million years to a fraction of millisecond) relative to the age of the Earth
that any atoms
of that element that may have been present when the Earth formed have long since completely decayed away. Hence they are only known on Earth as the product of nuclear reactors or particle accelerators.
The first synthetic element discovered was technetium, which filled a gap in the periodic table, and the discovery that there were no stable isotopes of technetium explained its absence on Earth: its 4.2 million year half-life meant that none remained from the time of formation of the Earth.