Painting over wallpaper?

im redoing my kitchen and it has a very ugly border and i cant get it off i have tried every thing i know of steam, water, water and fabric softener, chem........ so can i just put a few coats of primer over it and paint the wall???? also the wallpaper was put up by the builder and they cut corners and pretty much put it on the dry wall and the few spots i could get up took the drywall with it