WASHINGTON — In the space of a month, the Supreme Court has thrust itself into the center of American political life, agreeing to hear three major cases that could help determine which party controls the House of Representatives and whether President Obama wins a second term.

The court announced Monday that it would decide whether Arizona was entitled to impose tough anti-immigration measures over the Obama administration’s objections. The case joined a crowded docket that already included challenges to Mr. Obama’s signature legislative achievement, the 2010 health care overhaul law, and a momentous case on how Texas will conduct its elections.

The Texas case, which on Friday the court agreed to hear, could cause as many as four seats in the United States House of Representatives to change party control.

“It’s not just that these are big cases, but these are big cases that echo in the political arena,” said Nathaniel Persily, a professor of law and political science at Columbia University. “There is now a judicial forum for airing these political disputes.”