After watching film of their latest loss, the winless Cincinnati Bengals came away with a little bounce in their step and a sense that things are about to get a whole lot better.

Really.

The Bengals are the most upbeat of the NFL’s 0-for-’08 teams. A 26-23 loss to the Giants in overtime on Sunday convinced them that they’re headed in the right direction, even though their quarterback keeps getting his nose smashed and fans have started disowning them.

‘‘A lot of positives,’’ receiver Chad Ocho Cinco said. ‘‘A lot of good things. A couple of missed assignments across the board — you know, it was always one thing on every play, which it usually is when plays break down. But we had a lot of positives, man. We looked really good on film.’’

Not so good in the standings, of course. And that’s the problem.

At 0-3, the Bengals are tied with the Cleveland Browns for last place in the AFC North. They play Sunday in Cincinnati, a game that will give one of them a flicker of hope that they could still win a struggling division. The loser is in a whole lot of trouble, no matter how it looks on film.

‘‘I mean, what better week to get a win than against the Browns, you know?’’ Ocho Cinco said. ‘‘What better week?’’

The last game would have been the best one for the Bengals, who knew they had to beat the Giants to provide evidence that all is not lost. The offense looked decent for the first time all season, and the Bengals went to overtime before letting it slip away.

The offense was the main reason for optimism. Carson Palmer threw his first touchdown pass — he also got his second bloody nose of the year — as he threw for 286 yards against one of the NFL’s top defenses. His previous high for a game this season was 134 yards passing.

‘‘We were much sharper in a lot of areas in the passing game,’’ coach Marvin Lewis said. ‘‘We still have to work it and make sure we’re being exact and give (Palmer) the opportunity to deliver the ball where he thinks it ought to be delivered.’’

Palmer was sacked six times and got his nose bloodied for the second time — he broke it during the third preseason game. He’s been knocked around in all three games so far, a trend that is most troubling.

‘‘Once again, that ain’t a sight I want to see: My quarterback with a bloody nose,’’ offensive guard Bobbie Williams said. ‘‘It kind of makes me feel like I’m not really doing my job. That’s not a sight any guy — particularly offensive linemen — want to see.’’

The same offensive line allowed only 17 sacks last season, when Palmer threw for 4,131 yards. He’s been dropped nine times already, a mystifying development.

Told that a lot of fans are scratching their heads over the high number of sacks, Williams said, ‘‘Me, too. I’m about to scratch a patch in mine. But you know, we’re dealing with it. We have to get it done.’’

The Bengals haven’t lost their first three games since 2003, which was Lewis’ first season as head coach. They got back into the playoff race that year and were 8-6 before falling apart. They lost their last two games for an 8-8 finish.

The last time they lost their first four games was 2002, when they opened 0-7 and finished 2-14, the worst mark in franchise history. Coach Dick LeBeau was fired after that one.

One notable change in this 0-3: Ocho Cinco is biting his tongue.

In previous years, the receiver would have been insisting that Palmer get him the ball more often so the team can start winning. He has only nine catches for 99 yards in three games — numbers that would have brought an eruption in the past.

‘‘I’d (have) been here like a little pit bull, fussing,’’ Ocho Cinco said. ‘‘But no need, no need. I think right now, we’re fine. We might not be in a position that we really want to be in, but right now, we don’t need my problems being a distraction. After seeing what I saw on film, we got a lot of positive things.’’

The won-lost record happens not to be one of them.