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That’s a possibility that worries many Republicans. Polls, all conducted before the Tucson shootings, show Mrs. Palin to be the most divisive of the potential GOP candidates. Many Americans are solidly for or against her, and relatively few are undecided.

“Will Palin run?” is almost a parlor game in political circles. Wednesday’s video did little to settle it. Some politicians questioned why a presidential hopeful would take chances with phrases like “blood libel” at a time when many elected officials are trying to lower the rhetorical temperature.

Former Minnesota Gov. Tim Pawlenty, another possible Republican presidential candidate, told the New York Times that it’s wrong to blame politicians for the Tucson tragedy. As for the use of cross hairs to target House districts, which Mrs. Palin’s video suggested is commonplace, Mr. Pawlenty said, “It’s not a device I would have chosen.”

Some saw Mrs. Palin’s video as a sign she’s eager to challenge Mr. Obama. She twice referred to America as “exceptional.” That’s a favorite word of conservatives, who say the president refuses to acknowledge the nation’s well-earned prominence.

Some Republicans doubt that Mrs. Palin and her small group of confidants spend a lot of time in deep, strategic thinking. She seems to follow her instincts, they say, which have helped propel her to remarkable amounts of fame and wealth — starting, of course, when Sen. John McCain, the 2008 Republican presidential candidate, made her his running mate.

Many Democrats think Mrs. Palin is much better at making money and gossipy headlines than in assembling the kind of political operation that can carry her to the White House.

“Every time she pops off, she excites her narrowing band of partisans and probably makes herself more money, but she further alienates everyone else,” said Democratic consultant Jim Jordan, a veteran of presidential campaigns.

Historically, voters in Iowa and New Hampshire insist on questioning presidential hopefuls in small and frequent gatherings. That tradition might force Mrs. Palin to emerge from her cocoon. New Jersey Gov. Chris Christie, a Republican, told New York Times editors Wednesday that MRs. Palin will “never be president” if she continually avoids unscripted and possibly adversarial exchanges with reporters and the public.

Some campaign veterans, however, think Mrs. Palin may be able to use rapidly expanding social media outlets to reach and inspire primary voters in novel ways.

“She’s a very savvy practitioner of new media,” Mr. Smith said. A candidate probably cannot win the Iowa and New Hampshire Republican contests entirely with Facebook, Twitter and similar outlets, he said, “but you can do an awful lot.”