You never have to change your oil at 3000 miles, that's just hype from oil companies to make more money. Regular dino oil is good for about 8000 miles. Synthetic is good for over 10,000.
The oil filter is the weak link when it comes to the whole oil change thing. It supposedly becomes clogged or what not. With synthetic, if you wanted to push it, you could keep it in there for a long time, like 15,000 miles and change the filter only half way. Top off what ever spills out.
Oil companies want money, that's why they push the whole 3000 mile oil change deal. The owners manual says 5000 if certain conditions exsist, and 7500 if they don't, and that is for dino oil!
To keep the warranty valid I would change oil at every 5000 with synthetic. You could even wait to 7500. Then when you are out of warranty change it every 10,000 miles on synthetic. Save yourself the money. The engine will be fine.
This has been MHO...