Why are synthetic elements unstable?

Protons are positively charged, and neutrons are neutral, so large nuclei are highly positively charged. A postively charged sphere will energetically prefer to break up into two separate charged droplets which move far apart, this reduces the electrostatic energy, since the electrostatic field does work during this process.

This thing, spontaneous fission, is usually phase-space unlikely, since you need to have a large chunk of the nucleus tunnel away from another large chunk, and it's unlikely for all those particles to tunnel out together. But at large atomic numbers, you are unstable even just to shooting out an alpha-particle, and this doesn't require a conspiracy, so large Z nuclei are alpha unstable, usually with long half-lives.

The positive charge on nuclei puts a limit to the stable ones. The reason is simply that the electrostatic force is long range, while the cohesion force is short range. The same phenomenon causes the instability of water droplets, so that if you charge one up, it will break into a fine mist. The cohesion of the droplets is local, while the electrostatic repulsion is long range.

The scale at which you get a fission instability directly can be estimated from surface-tension considerations. If you break a sphere into two adjacent spheres of same total volume, the radius is reduced by the cube-root of two, so that the surface area is decreases by the square of this, and you multiply by 2 (since there are two spheres) so the net factor is the cube-root of 2, which is around 1.3. So the extra surface tension energy is increased by a factor of 1.3, or 30%.

But in separating the two spheres, you have taken one ball of charge, with an energy of $Q^2\over R$ and separated it into two adjacent balls of reduced radius and half the charge. Adding up the electrostatic energy, it is about 80% of the original electrostatic energy in the single sphere.

So spontaneous droplet fission will happen when you have a charged ball for which 30% of the surface tension energy is less than 20% of the charge energy. Since charge goes up almost as the volume (not quite, but close) while the surface tension goes up as the area, there is a crossover, and charged droplets will spontaneously separate when they are too big.

The surface tension can be found from the binding energy curve of nuclei, and these simple considerations limit stable nuclear size to about that of Uranium. The U nucleus can spontaneously fission at an extremely low rate, but the transuranics become progressively more unstable because their electrostatic energy is increasing as the volume to a power greater than 2/3, while their surface tension energy is increasing as the surface area, which grows as the 2/3 power of the volume.

These considerations, in much more sophisticated form, are due to Niels Bohr in the seminal liquid drop model of the 1940s. This model explained the nuclear binding energy curve quantitatively, and accounted well for fission phenomena. The only major thing left out of this was the shell model and magic numbers, which was supplied by Mayer.

I'd like to answer in a different direction: if one of them were stable we would have found it in nature, so it is not sythetic. Also the (now) synthetic elements were generated in the development of the universe, but they decayed because they were unstable.