How to test aluminum polymer capacitor for failure?
Technicians often use an ESR (Equivalent Series Resistance) meter, which works in-circuit and will find open parts. 

You can usually test in-circuit for shorts with an ohmmeter. 

Testing is done with the power off and after discharging the capacitor in both cases, of course. 

One crude test you can do if the cap is not shorted (to check for "open" or high ESR) is to parallel it (tack it on with a soldering iron) with a known-good capacitor and see if the circuit starts working. Only do this if you understand enough about the circuit to be sure it won't hurt anything.