Synthetic oil usually can go up to 10,000 to 15,000 miles before an oil change is actually needed but I would never let the oil change go past 15,000 miles.
Oil is the lifeblood of your engine and when it gets dirty and contaminated the oil cannot lubricate your engine and engine bearings properly.
This then leads to wear on your engine parts including the bearings which can then lead to a seized engine or your engine throwing a rod or having a spun rod bearing.
If I were running synthetic oil in my engines I would still change the oil filter after 3,000 to 5,000 miles even if I didn't change the oil at that time.
I always change oil filters though when changing oil regardless if the oil is regular oil or synthetic oil.