Dividing certain plants

So I have a single Pansy. Not really fussed with Pansies to be honest, although I do have a good little number of Violas, but I loved the really deep purple of the flowers and so I bought it last year. It's still in a small pot, but needs potting on really, or planting out (possibly in my border when I actually get round to finishing it). I was wondering if it's possible to divide it and have at least two plants from it? And then maybe divide it again next year and so on? It is a single plant and I can't really see a way to divide it without damaging it : /
I also have a certain alpine who's name I don't know. I love it, and I especially love it's flowers, which are out now and look like spires of teeny-weeny purple Orchid-type flowers. So tiny, but really beautiful. It's leaves are a rounded, purple-ish green and it's wanting to spread it's-self outwards with long, spreading steams. Not sure if that helps in any ID at all? If so, could I also divide this plant in some way(also a single)? I'd love to have more of them.
I'm quite familiar with division of other plants. I'm on such a low budget that I'll grow from seeds and/or buy tiny plants, grow them on and then divide them in their second or third year to further my number. But never sure if I can get away with that tactic for everything?
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You can clone them through cuttings. I've not done it myself, but I'm sure there is a world of information about it here.. hopefully somdone will advise.
I reckon if you pot it on straight away it'll produce new growth quite quickly; if you then pinch the tips out of the new shoots the plant will become more bushy, AND you can use the little bits you remove as cuttings. If you google "pansy cuttings" it'll give you chapter and verse, I'm sure.
Thanks peoples : )
Is now a good time to take cuttings off the Pansy, with it being in flower? There are non-flowering stems.
The Alpine, yes I could take a photo, but it's tipping it down out there right now : / I'm not sure if they're runners or it's just the way the stems grow, but I was wondering if I could do that. Never don't it before though. Any general rules to go by, or just treat the stems as if they were cuttings?
Having done some research I reckon Logan is right about the layer cuttings. A website for a specialist pansy society suggests covering the long stems with about 4" depth of a half-and-half mix of peat and silver sand (I guess you could use multi-purpose compost, actually), doing this in July-August. (Make sure the growing tips are sticking out of the compost.) By September-October you should have a lot of roots coming from the buried stems, and can separate the rooted bits and pot them up.
Thanks Liriodendron.
Not sure the stems are long enough yet to bury that deep. Maybe I should wait and just pot on for now.
Sounds a good plan, Blue Dragon.
Thanks for that info Bilje, it may well be an Ajuga. And it is continuing to spread it's stems outwards quite a way from the main plant. I haven't seen any sign of roots forming on these runners, but I will be trying to see if I can encourage some.