How can I divide my composition into exactly three parts to the pixel?

All good advice above, but just an additional option to consider.

You can design everything in a 5760x1080 comp as discussed. Create Guides to visually define the vertical join lines in your comp.

But, if you're not doing any additional processing to the images before you render, you don't have to create additional comps for output.  You can do it straight from the Render Queue.

Send your main comp to the Render Queue as normal.  In the Output Module section, click the checkbox to enable Crop.  Then define the dimensions you wish to render in a pass.  In your case, you will need to have 3 output modules, ones for each video frame.  So the dimensions will be:

Output Module 1 (Left) :        Top-0, Left-0, Bottom-0, Right 3840

Output Module 2 (Centre) :    Top-0, Left-1920, Bottom-0, Right 1920

Output Module 3 (Right) :       Top-0, Left-3840, Bottom-0, Right 0

You can save these Output Modules as templates so that you don't need to rewrite the numbers every time.  Plus, you can stack multiple Output Modules in one Render Queue item, meaning it only renders the main comp once, then distributes the data to each individual Output Module in one pass.

I am looking to divide a 5760 x 1080 comp three ways,

create a composition call it Part 1 and make it 1920x1080. put the main composition 5760x1080 inside it.

drag it to the left hold shift and Ctrl to snap it to the left edge of the composition.

note the X position parameter. it is 2880. you need to move it 1920 each time.

duplicate part 1, open part 2. on the x parameter add after 2880 " -1920"

duplicate part 2, open part 3. repeat the process.

if you want a visual reference to your splits, then in the master composition add a solid layer, set is as a guide layer so it won't render (right click->guide layer) and add  the grid effect. set its anchor point to 0,0, size from: width slide, set it to 1920 and set the border to something you could see like this:

Do your work in a 5760 x 1080 comp.  Then make double-dog sure about how this animation will be played back -- as one huge file, or three 1920x1080 file.

Do you have delivery specs for this project?  What are they?