Here's my take:
He wants to span 17'. The load being supported is a finished floor load from a residence. So the design load per square foot should be around 40 PSF/live and 10 PSF/dead for a total of 50 PSF.
This beam is supporting floor joists that are perpendicular to it. One side spans 11'-9", the other 12'-3". The beam will have to support the load over half the span of each side (the remainder of the load is supported by the concrete block walls on which the floor joists bear (or are attached).
So our total floor area to be supported is 12' wide (half of 11'9"+half of 12'3") by 17' long, for a total square footage of 204 sq. ft. Multiply this by 50 psf = 10,200 lb load.
What I'm not sure of it if there is additional loading to be supported, because I'm not sure if there is a wall above that is holding up weight and if there is ceiling and roof loading or if the trusses that he mentioned actually span the distance and while they may set on the wall that sets on the floor area, the floor actually bears (essentially) no additional load.
So now we've got to see if the existing beam will hold the weight.
The span table I have works in 2' increments, so I'll average between 16' and 18' span.
A 2x10 will hold up about 1,000 lbs over 17'. It's a triple 2x10, so that's only going to support about 3,000 lbs.
If it were supported at the mid-point of the span, then the loads are cut in half. A 2x10 at 8' span supports 2,139 lbs, so a triple would hold 6,417 lbs (which is well over half of our load of 10,200 lbs).
In the flow chart of this project, if he can add a post (and a footer), that will be the cheapest way to support this.
If a clear span is required, I would say he would need to go with an engineered beam or a steel beam.
From an old chart that I have, there are many beams that will support 10,200 lbs over 17 ft. The two lightest that I could find are a standard I-beam, 10 x 4 3/4 (25.4 lbs/ft) and a wide-flange I-beam (10 x 5 1/4, 21 lbs/ft). There are a couple of 8" tall beams that would support it as well. Both of the beams listed are rated to support at leat 16 kips at 18' span.
In my opinion, this is more of a carpentry equation than an engineering calcution. I think that we are dealing with (more a less) a residential situation, so dragging engineers into is probably not necessary.
Obvious disclaimer: unless local codes require stamped or engineer-approved plans, etc. etc. etc. I'm fortunate that I live in an area without such codes for residences, so I have never had to deal with building permits, inspections, etc. so I cannot account for that.
At any rate, bottom line, I don't think the existing beam is capaple of supporting the load, long term. I don't think that a good answer can be manufactured from dimensional lumber being attached to it. I think the options are either:
1. a post on the existing beam
2. a steel beam in place of or underneath the existing beam
3. a pre-engineered wood beam in place of or underneath the existing beam
Good luck and take care.
Edit:
Sorry, I didn't retain the part about the overall span being 34' and (I'm assuming) you want to separate it into two 17' spans instead of three 12'6" spans. In that case, I would use a steel beam under or in place of the existing and supporting it with a column (unless you wan't to add more posts).
And I see that an inspection is not required, so no worries there. I think it's obvious that the best thing to do is take the advice of unknown people from an internet forum! (This is what I hate about typing a post - I can can't convey voice inflection. I can't even find a proper emoticon. What I'm trying to do is come across with a good-natured ribbing comment, and I'm afraid it may sound like a condescending jack@$$. So I'm just trying to be funny, not a jerk....although sometimes, I can do both).
Anyway, good luck and take (again).