U.S. Democratic presidential nominee Hillary Clinton answers a question from the audience at a "Family Town Hall" campaign stop in Haverford, Pennsylvania, Oct. 4, 2016.
U.S. Democratic presidential nominee Hillary Clinton answers a question from the audience at a "Family Town Hall" campaign stop in Haverford, Pennsylvania, Oct. 4, 2016.

A raft of new U.S. presidential campaign surveys shows that Democrat Hillary Clinton has regained an edge over Republican Donald Trump since their first debate, which many voters say they think she won.

The national polls mostly show Clinton, a former U.S. secretary of state, leading Trump, a brash real estate mogul seeking his first elected office, and two other candidates, Libertarian Gary Johnson and Green Party contender Jill Stein, by 4 or 5 percentage points in a four-way race. In head-to-head matches with Trump, Clinton's advantage is slightly bigger.

Other surveys in closely contested battleground election states, all conducted in the days after their face-to-face encounter a week ago, showed Clinton gaining ground on Trump.

U.S. presidential elections are not decided by the national popular vote but rather by results in the country's 50 states, with the most populous states having the biggest influence on the overall outcome in the decisive Electoral College vote.

Politico, a political news website, said Clinton, looking to become the first female U.S. president, is now ahead in seven of the 11 states where the outcome is in doubt, giving her a far clearer path to winning the 270-vote Electoral College majority over Trump.

Clinton is also ahead in many of the biggest states, where voters for years have opted for Democratic presidential candidates and are likely to vote for her in the Nov. 8 election.

Voters surveyed

Political surveys showed voters, by about a 2-to-1 margin, thought Clinton won the first of three debates with Trump. She kept him on the defensive, attacking him for defying four decades of presidential campaign tradition by not releasing his U.S. income tax returns, his long history of slurs aimed at women, and his yearslong campaign to try to prove the debunked claim that President Barack Obama was born in Kenya and is not a U.S. citizen.

Venezuelan-born former Miss Universe Alicia Machad
FILE - Venezuelan-born former Miss Universe Alicia Machado presents an award at the Metropolitan Fashion Week's Closing Gala & Awards Show at Warner Brothers Studio.

Clinton also attacked Trump's disparaging remarks about the weight a Miss Universe gained in the year after she won the beauty pageant that Trump once owned, and he continued to mock Alicia Machado in Twitter comments he posted in the days after the debate.

Then, recently, The New York Times published three pages from Trump's 1995 state tax returns showing that he declared $916 million in business losses that year from failing casinos and other ventures. It was a deduction so large that it could have allowed the real estate mogul to legally avoid paying U.S. income taxes for up to 18 years.

Clinton derided his business acumen Monday, asking at a rally in the Midwestern state of Ohio, "What kind of genius loses a billion dollars in a single year?"

She contended that his legal tax avoidance "was taking from America with both hands and leaving the rest of us with the bill."

'Legally' used tax laws

Trump rebuffed the attack, calling his use of the country's tax laws the right of any taxpayer to write off business losses against income brilliant on his part.

U.S. Republican presidential nominee Donald Trump
Republican presidential nominee Donald Trump meets with energy executives during a campaign stop in Denver, Colorado, Oct. 4, 2016.

“As a businessman and real estate developer, I have legally used the tax laws to my benefit and to the benefit of my company, my investors and my employees. I mean, honestly, I have brilliantly — I have brilliantly used those laws,” Trump told supporters in the western state of Colorado. “I have often said on the campaign trail that I have a fiduciary responsibility to pay no more tax than is legally required, like anybody else, or put another way: to pay as little tax as legally possible. And I must tell you, I hate the way they spend our tax dollars.”

He added, “That did not happen by chance or luck. It happened by action and talent. Lot of talent. I was able to use the tax laws of this country and my business acumen to dig out of the real estate mess — you would call it a depression — when few others were able to do what I did.”

Trump, who lost millions of dollars in building casinos in Atlantic City, New Jersey, along the Atlantic Ocean coastline, said, “When the chips are down is when I’m at my very best. They said I was finished. Everybody said I was done. I am still here."

Vice presidential candidates

Democratic vice presidential candidate Tim Kaine (
Democratic vice presidential candidate Tim Kaine (L) and Republican vice presidential candidate Mike Pence

The focus Tuesday is on the vice-presidential debate between Trump's and Clinton's running mates, Indiana Republican Governor Mike Pence and Virginia Democratic Senator Tim Kaine. While debating their own credentials, both will try to make the case for the standard bearers, Trump and Clinton, five weeks ahead of Election Day, to defend their policies in the race to take over the White House when Obama leaves office in January.

Trump was campaigning Tuesday in the western state of Arizona, which usually has voted for Republican presidential candidates, but one where Clinton has edged close to him. Clinton campaigned in the suburbs of the large eastern city of Philadelphia, Pennsylvania, with her daughter, Chelsea, and actress Elizabeth Banks, trying to woo women who have voted for Republicans in the past.

Taking a question from a teenage girl, Clinton talked about concerns that young women have about their body image. She said Trump had insulted Machado and that his comments about women over the years presented "a new level of difficulty and meanness."

Clinton said of Machado winning the Miss Universe contest, "I mean, how do you get more acclaimed than that? But it wasn't good enough."

The Democratic candidate said parents, teachers, men and women have to "stand up against" cultural demands that only a woman's looks matter, not her mind or heart.

At another campaign appearance, Clinton criticized Trump for seeming to suggest that veterans who suffer from PTSD (post-traumatic stress disorder) might not be as strong as those who don't.

She said the post-traumatic stress still carries a stigma, adding it didn't help "when a man asking to be our next commander-in-chief stands up and says post-traumatic stress isn't a problem if you are strong."

Trump made the reference Monday as he discussed his commitment to improving mental health services for veterans.

Clinton and Trump are set for their second debate next Sunday, a town hall-style format in which undecided voters will have a chance to ask them questions.