Protests raged after Americans elected Donald Trump president. Another massive protest is planned for inauguration weekend. Online, his opponents express their dread that he could be "normalized."

Before the election, more Americans believed Trump's opponent had the better temperament and ability to serve as president.

And, yet, half of Americans now believe Trump will do a good job as president.

That's one finding from a new CNN poll that captures Americans' feelings toward the president-elect.

(And before we go forward, yes, there is reason to take polls with a few grains of salt, given Trump's surprise win on Nov. 8. Then again, Clinton is winning the popular vote still, by 1.7 million votes as of publication time, and is expected to inch up higher. Clinton is also ahead by 1.3 percentage points, and the final RealClearPolitics polling average showed Clinton ahead by 3 points, just 1.7 points off.)

Taken together, this and other post-election polls suggest Americans are cautiously optimistic about a Trump presidency. One poll shows his favorability rating climbing. Another shows that even a majority of Clinton supporters want to "give him a chance." However, even some of his own voters have reservations about him (and particularly his children's influence). And, of course, the nation remains heavily ideologically divided.

Half of Americans think Trump will be a good president

Fifty-three percent of American adults say they believe Trump will do a "very good" or "fairly good" job as president, according to the CNN poll out Tuesday.

That's fairly good news for a man who had some ugly polling numbers before the election. As of late October, a CNN poll showed that only 40 percent of registered voters thought he could "better handle the responsibilities of commander-in-chief" and one-third thought he had the "temperament" to serve. And only 4 in 10 registered voters viewed Trump favorably (to be clear, that's a smaller population than "American adults").

On the other hand, this number isn't all that impressive compared to Trump's predecessor. Just after the 2008 election, fully three-quarters of Americans believed Obama would be a very or fairly good president.

A majority of voters — 56 percent — say they believe Trump will have a "successful" first term as president, according to a Pew Research Center poll released this week.

Once again, it's perhaps an encouraging finding for Team Trump, but it's not quite as good as Obama's 2008 marks (two-thirds thought he would be successful) — and he had no glide path when it came to legislating.

Importantly, "successful" could mean any number of things to different people. Does it imply policy outcomes that the voter considers good? That Trump could be successful in pushing his policies through Congress? Or something else?

Another poll suggests voters are slowly warming to Trump. A new Morning Consult poll shows Trump's favorability ratings have climbed: right now, Americans are evenly split, with 46 percent viewing him favorably and 46 unfavorably. That's up from a 37 favorable, 61 unfavorable split before the election.

Considering the continuing outrage over his policy proposals and appointments (more on this below), that favorability rating could hit a ceiling soon. But considering all those other times the world tried to guess where the Trump ceiling was, hazarding a guess here is probably a bad idea.

Voters don't want leaders to cooperate — except they kind of do

Yes, more people seem to see Trump favorably than did prior to the election. However, many Americans remain strongly opposed to him. While half of Americans are "hopeful" or "happy" about his election, half are also "uneasy." Four-in-10 are also sad, and 4 in 10 are scared, according to Pew.

So it's no surprise that Clinton voters want their party to be obstructionist. Nearly two-thirds say Democrats should "stand up to Trump on issues important to Democrats — even if it means less gets done." (Interestingly, 15 percent of Trump voters said the same thing.)

Several of Trump's post-election actions have worried his opponents: his appointment of Stephen Bannon as a top adviser is one — Bannon served as chairman at Breitbart News, which has published misogynistic articles, and which Bannon called a platform for the alt-right, a movement that is closely associated with white nationalism.

Trump opponents worry about the fact that his campaign has gained the approval of white-nationalist groups, as well as the possibility of a Muslim registry (RNC Chairman Reince Priebus told NBC's Meet the Press last weekend, "Look, I'm not going to rule out anything." But he then added, "We're not going to have a registry based on religion" and said there needed to be more vetting to keep "radical folks" out.)

It's not that Clinton voters are uniquely unwilling to compromise here; only one-quarter of Trump voters say Trump should pick Democrats to serve in his administration (half of Obama's voters said so of Republicans in 2008).

In many of Pew's questions, it's clear that voters are heavily divided. Back then, 40 percent of McCain voters thought Obama would have a successful first term — less than half as many Clinton voters say so of Trump.

In light of all that, here's what's jarring: Despite their polarization, Americans still hope leaders can work together — 55 percent of Trump voters want him to "work with Democrats to get things done, even if it means disappointing supporters." And 58 percent of Clinton supporters say they are "willing to give Trump a chance to see how he governs."

So maybe Americans really underneath it all are OK with their own sides compromising a bit if it means a more effective government.

Or they're just hopeful that the other side will cave first.

Voters concerned about conflicts of interest

This is Trump's "honeymoon phase," as Morning Consult co-founder Kyle Dropp said of Trump's higher favorability marks. But even mid-honeymoon, voters have reservations about how much Trump will (or won't) keep his family business and politics separate.

A clear majority of Americans (59 percent) believe that Trump handing over his business to his children "does not go far enough" toward preventing conflicts of interest.

This includes some people who in general support him: nearly 4 in 10 conservatives and one-quarter of Republicans agreed that giving his children control of the Trump Organization isn't enough.

Recently, Trump claimed his businesses would be placed in a "blind trust" led by his children. However, several outlets have argued that that's not a terribly "blind" setup.

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