Sarah Palin was a smart hire for Fox News Channel, judging by the ratings for her debut on “The O’Reilly Factor” on Tuesday. The former Alaska governor’s first appearance as an analyst drew 3.954 million viewers — more than everything else on other cable new shows combined in the 8 p.m. hour. CNN’s “Campbell Brown” placed second with 999,000 total viewers, MSNBC’s “Countdown with Keith Olbermann” placed third with 906,000 total viewers and HLN’s “Nancy Grace” brought up the rear with 704,000. Compared to host Bill O’Reilly‘s ratings the night before, the Palin push was worth a 22 percent rise in ratings, or more than 1 million more viewers including the repeat of O’Reilly’s show at 11 p.m. The former vice presidential candidate, who appeared yesterday for a full hour on Fox News with Glenn Beck, will be on “Oprah” today with James Cameron and Lady Gaga.