Where do polar bears live, & where do polar bears go?

Firstly they don't live in Antarctica with penguins.
They live in the Arctic. That's the big icy bit around the North Pole, duh.

Well actually to be more precise they mostly live around the land masses around the edge of the arctic polar basin, on the sea ice near coastlines and islands. They can become stranded on land when the sea ice melts in summer.

They are not normally found further North than 82 North Latitude since food gets a bit scarce that far north, but there have been rare reports of them as far north as the North Pole

How far South Polar Bears venture is usually dependent upon how much sea ice forms during winter. Polar Bears very much prefer to travel across sea ice, and, during heavy ice years they have been found as far south as the Pibilof Islands, Kamchatka, Newfoundland and Iceland. James Bay is their most southerly year round dwelling.

Polar bears can do a bit of travelling in their lifetime. Their home range can be as much as 350,000 square km, but if there is plenty of food in the area it might be as 'little' as 50,000 square km.
This territory is not marked by the bears in any way as they aren't a very territorial creature, they don't defend their territory from other bears.
They do however spend much the same time of year in the same area as they were in the year before.
Walking they can cover a distance of 30km a day for several days at a time.
They can swim for several hours at a time, and have been tracked for over 100km continuously swimming

When females are pregnant they spent autumn and winter in dens on land. When the female is pregnant she will in october, (or sometimes as early as August) seek a den.

The dens are usually dug on snow drifts on southerly facing slopes. While most of them are dug within the snow, some may be dug into dirt on land to become covered in snow later.

The den is usually a single chamber, slightly elevated from a short entrance tunnel. About 2 meters long and 1.5m and 1m high. A ventilation hole through the chamber is maintained so as to provide fresh air. The den is warmer that the outside air temperature due to the insulation of snow and the body heat within the den.

Dens can be located within a range from sea level to over 500m above sea level. Most are located on land within 16km from the coast, but in some areas they can be over 100km inland from the coast, or sometimes located on the sea ice.

The mother & her cubs will emerge from the den in late march or april.