Etymology
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algebraic (adj.)
1660s, from algebra + -ic. Earlier was algebraical (1570s); algebraically.
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algebra (n.)

"formal mathematics; the analysis of equations; the art of reasoning about quantitative relations by the aid of a compact and highly systematized notation," 1550s, from Medieval Latin algebra, from Arabic "al-mukhtasar fi hisab al-jabr wa al-muqabala" ("the compendium on calculation by restoring and balancing"), the title of the famous 9c. treatise on equations by Baghdad mathematician Abu Ja'far Muhammad ibn Musa al-Khwarizmi. Arabic al jabr ("in vulgar pronunciation, al-jebr" [Klein]) "reunion of broken parts" (reducing fractions to integers in computation) was one of the two preparatory steps to solving algebraic equations; it is from Arabic jabara "reintegrate, reunite, consolidate." Al-Khwarizmi's book (translated into Latin in 12c.) also introduced Arabic numerals to the West. John Dee (16c.) calls it algiebar and almachabel. The accent shifted 17c. from second syllable to first.

The same word was used in English 15c.-16c. to mean "bone-setting," as was Medieval Latin algebra, a usage picked up probably from Arab medical men in Spain.

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al- 
in words from Arabic (or assumed to be), it is the definite article "the." Sometimes rendered in English as el-. Often assimilated to following consonants (as-, az-, ar-, am-, an-, etc.). Examples include almanac, alchemy, alcohol, algebra.
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algorithm (n.)
1690s, "Arabic system of computation," from French algorithme, refashioned (under mistaken connection with Greek arithmos "number") from Old French algorisme "the Arabic numeral system" (13c.), from Medieval Latin algorismus, a mangled transliteration of Arabic al-Khwarizmi "native of Khwarazm" (modern Khiva in Uzbekistan), surname of the mathematician whose works introduced sophisticated mathematics to the West (see algebra). The earlier form in Middle English was algorism (early 13c.), from Old French. Meaning broadened to any method of computation; from mid-20c. especially with reference to computing.
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idempotent (n.)
in algebra, quantity which multiplied by itself gives itself, 1870, from Latin idem "the same, identical with" (see idem) + potentem "powerful" (see potent).
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polynomial 

1670s (n.), in algebra, "an expression consisting of many terms;" 1704 (adj.), "containing many names or terms;" irregularly formed from poly- + stem of binomial. By 1885 as "a technical name consisting of more than two terms."

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nth 

by 1717, in algebra textbooks, in phrase to the nth, a mathematical term indicating an indefinite number, in which n is an abbreviation for (whole) number (n.). Figurative (non-mathematical) use is by 1852.

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exponent (n.)

1706, from Latin exponentem (nominative exponens), present participle of exponere "put forth" (see expound). Earliest use is the mathematical one (said to have been introduced in algebra by Descartes) for the symbol placed above and to the right of another to indicate by what power the base number is to be raised. The sense of "one who expounds" is by 1812. As an adjective, "exemplifying, explicating," from 1580s.

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