The Republican wave that washed over much of the nation Tuesday didn't completely bypass Minnesota. It was strong enough — particularly in Greater Minnesota — to flip the changeable Minnesota House back into Republican hands, ending two years of DFL grip on both chambers of the Legislature. (The Senate was not on Tuesday's ballot and remains in DFL hands.)

That result provided one big bright spot for Minnesota Republicans on Wednesday, after an election in which DFLers won every statewide race and re-elected all five DFL members of the U.S. House. It also sobered the DFLers, who know how difficult governance in St. Paul can be when power is divided, as it was from 1990 through 2012.

That's a valid concern. Yet we welcome the resurgence of a state GOP that was on the mat just two years ago, mired in debt and plagued with internal division. Minnesota needs at least two strong parties — and after this election, it's plain that it does not have three. The Independence Party that elected Gov. Jesse Ventura in 1998 lost its major-party status Tuesday, diminishing its chances of being a competitive force in the next several elections.

We also believe that despite the bad example set in hyperpartisan Washington, Minnesota Republicans and DFLers are capable of collaborating sufficiently to produce positive results.

The new House GOP majority would do well to learn from the mistakes of new majorities of both parties in previous years. That would include the last time a DFL governor and Senate and a newly elected Republican House were in charge, in 1985-86, when a Republican push for oversized tax cuts and deep reductions in welfare assistance for low-income single parents made for two rocky legislative sessions and a short tenure for the GOP. DFLers were back in charge of the House after the 1986 election.

The lessons from that year and all of the years since when the House changed hands (1986, 1998, 2006, 2010 and 2012): Don't overreach. Don't aim to please your own base in a way that neglects — or offends — the rest of the state. Don't stake out positions with the sole aim of putting the opposite party in a bad light, lest that light reflect back on you. Win-win solutions with the Senate and DFL Gov. Mark Dayton should be the new House majority's goal.

Such solutions ought to be achievable in 2015, particularly in these areas:

• Education tenure reform: A Republican House should bring improved chances for a bill to allow teacher performance as well as seniority to be considered when public school staffing adjustments are made. Dayton vetoed a 2012 bill with bipartisan backing that would have made that change, saying he wanted a new teacher evaluation system implemented first. That system is up and running this year. Given the new lineup at the Capitol, Dayton would do well to summon his political allies at Education Minnesota, the teachers' union, and invite them to help craft a compromise that helps retain high-performing young teachers.

• Transportation: Republicans deserve a chance to make good on what many of them said during the campaign — namely, that they favor more investment in roads and bridges in 2015. Minnesotans should hold them to that promise, and not settle for shifting a few million dollars on MnDOT's balance sheet when it will take an additional $20 billion over the next 20 years just to maintain the transportation status quo. For their part, DFLers should be willing to settle for an incremental approach to shoring up transportation financing.

Tuesday's election makes more state funding for transit, especially light rail in the metro area, much harder to come by. The GOP is notoriously rail averse and, after Tuesday's GOP victories in 10 outstate House districts, it will also resist spending on metro-only improvements. A quest for metro-based alternatives to state funding for new rail projects ought to begin in earnest now, and business backers of the proposed Southwest light-rail line should play a leading role in that effort.

• Corporate tax relief: "Tax" is a fighting word at the Capitol. But the new House majority may be surprised by how willing Dayton and DFL senators are to ease the tax burden on small and midsize businesses, especially if the state's balance sheet stays strong. Key Senate DFLers and Dayton have signaled a desire to cut corporate rates. The House, which must originate tax bills, should hone in on those signals.

• Sex offenders: Divided state government may be the only kind that can handle the political and legal hot potato that the Minnesota Sex Offenders Program has become. Next year, both parties will share responsibility for conforming that program to the U.S. Constitution while still protecting the public from miscreants. To their credit, Senate Republicans have already contributed to one possible framework for reform. House Republicans dodged the issue in 2014. They can't easily get away with that next year.

Doing those things — plus setting a new state budget without resorting to a government shutdown — would show Minnesotans that divided state government can be trusted. And shoring up trust in government ought to be on every Minnesota elected official's agenda — next year and every year.