Say the model of the motherboard. I guess it's an Asus, as the capacitor drawing on the motherboard has the negative side of the circle on the positive side. The model is usually written somewhere between expansion slots, and sometimes there's a revision number in a corner of the circuit board.
Grab a digital multimeter and test the power supply, make sure it outputs 5vSB (stand-by) when shut down.
Then put the meter in continuity mode and test all fuses that are on the motherboard. If you don't know what a fuse is, a good example of one is below that 10 pin USB 2 header, that tiny component with a p and a 15 written on it - which probably translates to "polyfuse, 1.5A trip point" or something like that.
If you don't get continuity over a fuse that can be a sign that something after that fuse died, shorted out or consumed too much power and you should look in that direction for a fault.
Most likely, the motherboard is dead because a mosfet is dead, or a voltage regulator is dead.
You have a bunch of mosfets under the heatsink near the CPU, you should have some mosfets by the 24pin connector, there's some mosfets in the picture you uploaded, those Nikos po1038 in the left corner of the heatsink (p06 and p07, p09,p10)
Those you can test with the meter in diode mode. You need to figure out if it's n-channel mosfet (most often) or p-channel mosfet and which pin or pins are the gate, which are drain and which are source - drain and source are usually connected to wide thick traces, gate is usually a lonely pin with a thin trace.
This gives a decent explanation on how to test a mosfet:
This shorter one made by a EEVBlog member says something similar :
Basically, on the motherboard you should not have a short between the drain and source pins of a mosfet, or between gate and the other pins.
Here's an example of a motherboard with a shorted mosfet (he used continuity mode to find a short between the leads of the mosfet) :
Jump to 2:45 if you're impatient:
And to answer the original question, those polymer capacitors are made by Apaq and they're most likely fine. It's very very rare they die or fail.