polieren
to polish
To make a surface smooth and shiny by rubbing it, often using a special product.
Er verbrachte den ganzen Nachmittag damit, sein altes Auto zu polieren.
He spent the whole afternoon polishing his old car.

to polish
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to polish
To make a surface smooth and shiny by rubbing it, often using a special product.
Er verbrachte den ganzen Nachmittag damit, sein altes Auto zu polieren.
He spent the whole afternoon polishing his old car.

You can recognize the meaning of 'polieren' (to polish) from the object: if you polish a physical thing like furniture or silver, it is meant literally. If you polish something abstract like an image, it should be understood figuratively.
Zuerst poliere ich die alten Schuhe, dann kümmere ich mich um mein schlechtes Image.
First I'll polish the old shoes, then I'll take care of my bad image.
In its literal meaning, 'polieren' (to polish) is often used for surfaces that are supposed to shine. Typical phrases are: 'Schuhe polieren' (to polish shoes), 'das Auto polieren' (to polish the car), 'Silber polieren' (to polish silver), or 'Möbel polieren' (to polish furniture).
Jeden Samstag poliert er sein Besteck aus Silber.
Every Saturday he polishes his silver cutlery.
In the figurative sense, 'polieren' (to polish) is used for improving something abstract. Common expressions are 'sein Image polieren' (to polish one's image), 'seinen Ruf polieren' (to polish one's reputation), or 'seine Fähigkeiten polieren' (to polish one's skills).
Mit ehrenamtlicher Arbeit versucht der Konzern, seinen Ruf zu polieren.
Through volunteer work, the corporation is trying to polish its reputation.
The expression 'jemandem die Fresse polieren' (to smash someone's face in) is very vulgar colloquial speech. It expresses an aggressive threat and should only be used or interpreted with caution.
Im Streit drohte er: „Pass auf, sonst poliere ich dir die Fresse!“
During the argument he threatened: 'Watch it, or I'll smash your face in!'
In the idiom 'jemandem die Fresse polieren' (to smash someone's face in), the person who is to be hit is always in the dative.
Er hat ihm gedroht, ihm die Fresse zu polieren.
He threatened him that he would smash his face in.
to improve
'verbessern' and 'polieren' can both mean that something becomes better or more refined. 'verbessern' is the general word when the quality, performance, or condition of something is meant to improve. 'polieren' literally means to make a surface shiny. Figuratively, it is used when something already exists and is made even more refined, tidy, or convincing, for example a text, a performance, or a résumé. So 'verbessern' fits almost any kind of making something better. 'polieren' is more appropriate when you mean the final polish or finishing touch.

Tom poliert vor der Bewerbung noch seinen Lebenslauf.
Before applying for the job, Tom polishes up his résumé a bit more.
Nach dem Gespräch verbessert das Team den Plan für das Projekt.
After the discussion, the team improves the plan for the project.