In one of only three League One games to go ahead, neither side looked like breaking the deadlock.
Pools, now five games unbeaten under new boss Neale Cooper, had two early openings. First in a frantic goalmouth scramble, the ball would not fall for either Sam Collins or Andy Monkhouse to nail a clear strike on goal.
Then Gary Liddle swept over an early cross from the right and teenager Luke James, watched by a number of scouts from higher division clubs, flicked a header at goal that was plucked out of the air by Bournemouth goalkeeper Darryl Flahaven.
But as the half went on, the Cherries started to get on top. Wesley Thomas should have scored in the 38th minute, but he blazed high over from 10 yards after a clever cut back from Marc Pugh.
Then Pools had Evan Horwood to thank for a stretched goalline clearance in the 44th minute after a header from Steve Cook looked like creeping in. Antony Sweeney prodded over from 10 yards in the second half, but further chances were scarce.