pol

This is less one root than a pile-up of look-alikes, and the honest thing is to separate the strands.

The real Latin thread is polīre, 'to polish.' Picture a craftsman rubbing a rough surface again and again until it gleams. That literal act survives directly in polish — you polish shoes, polish silver, and (figuratively) polish a draft until it shines. Latin also had interpolāre, 'to touch up, refurbish,' from the same family: to interpolate text once meant to slip new bits into an old manuscript, smoothing them in. Mathematicians borrowed it for filling in a value between two known points; extrapolate (extra- 'beyond' + the same stem) became its mirror — estimating a value beyond the known range. So polish, interpolate, and extrapolate form one small, coherent Latin 'smooth it over' family.

Now the impostors. polar and polarize have nothing to do with polishing. They come from Greek polos, the pivot or axis a sphere turns on — hence the Earth's poles, polar bears, and (by metaphor) polarized opinion: two camps at opposite ends of an axis.

monopoly hides yet another Greek word: polein, 'to sell.' mono- (one) + polein = the right of a single party to sell. That is why anti-monopoly and near-monopoly are about markets, not magnets or mirrors.

Finally necropolis: necro- (dead) + polis (city) = a 'city of the dead,' an ancient cemetery. polis is the Greek for city (also behind metropolis, police, politics) — again unrelated to polishing.

The takeaway for a learner: don't trust the spelling p-o-l to mean one thing. Ask which story you are in — smoothing (polīre), an axis (polos), selling (polein), or a city (polis).