Questions

$(1)$

Find formulas for a three parts Karatsuba's trick. Using MAPLE or AXIOM, for setting up these formulas, would be a good idea.

$(2)$

Is this three parts Karatsuba's trick theoretically better than the usual two parts one? That is, does this lead to a better time complexity upper bound?

You will need the so-called Master Theorem for analyzing the complexity of divide-and-conquer algorithms. See the document Master Theorem: Practice Problems and Solutions posted on our web site.