Here the goal is to reduce an arbitrary SAT problem to 3-SAT in polynomial time using the fewest number of clauses and variables. My question is motivated by curiosity. Less formally, I would like to know: "What is the 'most natural' reduction from SAT to 3-SAT?"
Now the reduction that I've always seen in text books goes something like this:
First take your instance of SAT and apply the Cook-Levin theorem to reduce it to circuit SAT.
Then you finish the job by the standard reduction of circuit SAT to 3-SAT by replacing gates with clauses.
While this works, the resulting 3-SAT clauses end up looking almost nothing like the SAT clauses you started with, due to the initial application of the Cook-Levin theorem.
Can anyone see how to do the reduction more directly, skipping the intermediate circuit step and going directly to 3-SAT? I would even be happy with a direct reduction in the special case of n-SAT.
(I would guess that there are some trade-offs between computation time and the size of the output. Clearly a degenerate -- though fortunately inadmissible unless P=NP -- solution would be to just solve the SAT problem, then emit a trivial 3-SAT instance...)
EDIT: Based on ratchet's answer it is clear now that the reduction to n-SAT is somewhat trivial (and that I really should have thought that one through a bit more carefully before posting). I'm leaving this question open for a bit in case someone knows the answer to the more general situation, otherwise I will simply accept ratchet's answer.