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1540s, "drive away," from Latin exterminatus, past participle of exterminare "drive out, expel, drive beyond boundaries," also, in Late Latin "destroy," from phrase ex termine "beyond the boundary," from ex- "out of" (see ex-) + termine, ablative of termen "boundary, limit, end" (see terminus).
Meaning "destroy utterly" is from 1640s in English, a sense found in equivalent words in French and in the Vulgate; earlier in this sense was extermine (mid-15c.). Related: Exterminated; exterminating.