The governor has a lower public approval rating than Richard M. Nixon had in his last month in the White House. The governor rarely appears before large gatherings because the booing tends to drown out his words. The governor has been abandoned - indeed, denounced - by most leaders of his own party.

But New Jersey Gov. Brendan T. Byrne, no matter what else his numerous enemies might say about him, cannot be called a quitter.

Ignoring the negative signals, he is busily campaigning for renomination in the Democratic gubernatorial primary election that will be held here next Tuesday.

And the governor just may win.

Byrne has stumbled onto an age old formula for success: divide and conquer. There are ten candidates seeking the Democratic nomination, and at least five had a solid chance to win.

Since there is no runoff here, a candidate with 25 per cent of the vote should win with nomination. Polls indicated Byrne may put together that much support from hard core liberals and voters near Atlantic City, who appreciate his backing for casino gambling in the resort.

A Byrne victory Tuesday would bode well for the Republican Party's chances to regain the State House. A Rutgers University poll showed that the average Republican by almost 20 percentage points in a statewide race. But if Byrne were the Democratic candidate, the poll projected a runaway Republican victory.

The governor seems to be a personification of Murphy's Law. Nearly everything he does goes wrong.

Byrne was one of the first governors to support Jimmy Carter in 1976. But when Carter campaigned in Newark last fall, Byrne bounded onto the platform, bringing a 10-minute chorus of boos from the audience. Byrne's relations with Carter have been icy ever since.

Like three previous governors, Byrne fought hard for a state income tax. Unlike the others, he won the fight.

But the tax, first collected this year, has been enormously unpopular. It has haunted Byrne throughout the campaign - even though all but one of his major opponents agree that it was essential.

Byrne fought for and won state financing of gubernatorial campaigns, so that candidates would not be indebted to fatcat contributors.

That hasn't worked either. The plan does not apply to primary campaigns, and this year the damage has already been done.

Like three previous governors, Byrne fought hard for a state income tax. Unlike the others, he won the fight.

But the tax, first collected this year, has been enormously unpopular. It has haunted Byrne throughout the campaign - even though all but one of his major opponents agree that it was essential.

Byrne fought for and won state financing of gubernatorial campaigns, so that candidates would not be indebted to fat cat contributors. That hasn't worked either. The plan does not apply to primary campaigns, and this year the damage has already been done.

The major Democratic hopefuls have raised about $4 million in the primary, most of it in large contributions. They have used it to buy time - at $1,000 or more per minute - on the New York and Philadelphia television stations that serve this state, one of only two in the nation without a VHF commercial station assigned to it.

Major Democratic candidates are Reps. James J. Florio and Robert A. Roe; a former state senator, Ralph DeRose; and Joseph Hoffman, who was fired from Byrne's cabinet after calling the governor "inept" in public. Each portrays himself as the party's only strong alternative to the governor.

Chief spender has been Roe, a five-term congressman who sponsors major public works bills. He has raised more than $500,000, much of it from construction firms, and has borrowed $300,000 more.

Since a relatively small number of votes will be needed for victory, the candidates are working for large turnouts in their particular areas of strength.

Florio, a second-term congressman from Camden, says he will garner a large plurality in the farm counties of southern New Jersey and pick up votes in Italian wards of the northern cities.

DeRose is counting on strong machine support in two northern counties that form one giant suburb sprawling westward from Manhattan. If the organizations can deliver, observers say, De Rose could win with no support from the rest of the state.

Roe and Hoffman both count on major labor backing.

The Republican primary pits two moderate state senators, Raymond Bateman and Thomas Kean, Bateman, who has won endorsements from most GOP leaders, is expected to defeat Kean, who has used a media campaign to lure rank-and-file Republicans.

Opponents in both parties deplore Byrne's warlike relations with the legislature and his failure to fill judicial vancancies. They say he cares mostly about the trappings of office, citing his three "official" visits to Britain and Asia and his frequent trips to conferences and celebrity athletic matches.

No one can say where Byrne's votes will come from, but everyone seems to think a victory is there somewhere.

"Head-to-head with Mickey Mouse, Byrne loses," says Hoffman, the former cabinet officer. "But with five of us splitting the vote, he could win."

"Everybody else says he's first in his own area, and I'm second," the governor chortled this week. "That adds up to I'm first overall."