Term for how politicians denigrate their opponents to win the support of the populace

You could say they are using ad hominem arguments, or engaging in ad hominem attacks. It generally means attacking the person rather than their arguments.

(edited after realizing Andy was looking for the word for denigrating, and not just overall lying)

You would say that they're scapegoating the opposition.

From Merriam-Webster's, a scapegoat in the figurative sense is:

1. one that bears the blame for others
2. one that is the object of irrational hostility

The transitive verb "to scapegoat" means "to make a scapegoat of someone".

They could be addressing Strawman arguments

This basically means they're deliberately presenting distorted, exaggerated versions of their opponents' arguments that are very easy to tear apart. For instance:

Senator Jones says that we should not fund the attack submarine program.

I disagree entirely. I can't understand why he wants to leave us defenseless like that.

Sound familiar?

Depending on the situation, many of these may apply :-)

The Synonym Finder - Page 662 Jerome Irving Rodale - 1978

lie

2

misrepresentation ... torturing; canard, rumor, hoax, forgery; monstrous lie, the big lie, mendacity, SI. whopper, SI. barefaced lie, SI. dirty lie, SA shameless lie; defamation, scandal, traducement, calumny, calumniation.

Jeff's "ad hominem" phrase is sometimes known – to footballers uncomfortable with Latin, perhaps – as "if you can't get the ball, get the man who's got it".

"Demagoguery" (augurar was first) has to be the market leader here. One could also work in a reference to "bread and circuses" or "panem et circenses" It's in the accusative case because "give'em" is understood as preceding). Though this is not about trashing the oppo.

Shakespeare probably has some good terms for flattering the mob in "Julius Caesar" and "Coriolanus", I suggest you riffle the pages.

I agree that "polemic" is misplaced. It does carry an overtone of being "over the top", though, and the root ("war") reminds me of the "Philippic", or oration calling for national resistance to an outside threat.

Might I also suggest "calumniation" and "scapegoating"? Then you have "black propaganda" against your opponents. When this was done against the Jews in the Middle Ages, we call it "the Blood Libel".

The common term is mudslinging, which is a noun:

the use of insults and accusations, especially unjust ones, with the aim of damaging the reputation of an opponent.

politicians seem to rely on mud-slinging instead of argument

And a verb:

make malicious or scandalous allegations about an opponent with the aim of damaging their reputation.

it's difficult to see how any parliamentary opposition would voluntarily give up the opportunity to smear and mudsling

It's not specific to politics despite most of Lexico's examples being about politics.