Each of the two temporary structures (one in each room) will need to mimic the section of wall that is being removed. So if the jack poles are resting on a single bottom-plate and are holding up a double top-plate, then jack poles 16" on center are called for. If something more substantial than the double top plate and single bottom plate are used, then fewer jack poles are required. In the extreme, it'd take only two jack poles placed 6ft apart to hold up their own temporary 4x10 header. The first floor's floor will need similar support in the basement.
I'd build the temporary support structure using lumber that I could use again somewhere else.
All that is assuming the ceiling joists are perpendicular to the wall in question and that the support structures are placed close to the wall in question.
If instead the ceiling joists are parallel to the wall in question, then there must certainly be a 2nd flr wall that is on top of the wall in question, in which case all the temporary support structures will need to be installed up on the second floor.
Personally, though, my choice would be to do it like this, without shoring up ...
0) pick through the pile at the lumber yard for straight 2x10s
1) open the wall on both sides
2) install the king and jack studs
3) notch the top of the studs on one side to let in one 2x10
4) install the that 2x10 onto the jack studs
5) shim between the cut tops of the studs and 2x10
6) cut off the stud ears (produced by the notching) that are on the side
7) nail off the 9" wide rippings of 1/2" plywood filler
8) install the second 2x10 onto the jack studs
9) nail off the header
10) remove the shims
11) remove the old studs
Best Answer
Yes, but I'd add a few things...
Firstly, I've done what you're doing many times. It will work just fine. A framed structure is a system, and there's a lot more supporting those joists than you might think. Mentally step through everything that they're connected to and you'll realize that you can almost take the studs out without doing anything else, even without your inner wall.
Now, FreeMan makes a good point about the studs on the inner wall not aligning with the joists, but not quite like he thinks. Joists needn't align with studs below if there's a doubled top plate. That's where your inner wall is lacking.
Fortunately, a solution is simple. Slap three temporary studs against the inner wall, flat, directly under the joists that will be unsupported by the outer wall. They'll run from the concrete to the underside of the joist. Nail them to the inner wall at the top, middle, and bottom and carry on. Everything else is well supported by the foundation wall. The temporary studs rest on the slab, which in turn rests on the footing below (or very close to it). All good.