To cut and seal the ends of a synthetic rope you will need paper masking tape, scissors (EMT/medical shears are recommended especially to cut larger diameter ropes), and a lighter (a torch lighter may be preferable but not a necessity).
Please be careful when using this technique that you do not burn yourself on the flame or the melted fibers of the rope before they cool.
Wrap the paper masking tape around the rope very tightly.
Cut the rope in the middle of the masking tape to the desired length.
Using the lighter, burn the end of the rope (the tape will probably catch fire, just shake it out).
Allow the rope to cool entirely.
Remove the tape to expose the new end of your synthetic rope.
For a video tutorial, please check out the following video: