MINNEAPOLIS — Minnesota began what is expected to become the broadest shutdown of state services in its history early Friday, after Republicans and Democrats there failed to agree on how to solve the state’s budget woes in time for the new fiscal year.

And so, on the eve of a holiday weekend, residents were likely to find the state’s parks, historical sites and the Minnesota Zoo closed, hunting and fishing licenses no longer being issued, and that state’s lottery system and racetracks unavailable. Minnesota’s 84 major rest areas along highways were closed. Thousands of state employees were expected to be sent home without pay, and contractors were to be told to walk away from hundreds of road construction projects already underway.

Since early this year, politicians in St. Paul have been locked in a battle over how to work out an expected a $5 billion budget deficit under a divided government. Republicans, who took control of both chambers of the Legislature after elections last year, called for cuts and reining in spending to the $34 billion that the state expected to take in over the next two years. But Gov. Mark Dayton, a Democrat who was also elected in 2010, called for collecting more in income taxes from the very highest earners to spare cuts in services to the most vulnerable residents.

While private negotiations went on, day after day, and the Friday deadline approached, it seemed as though the argument had never really shifted much at all.