Polynomials have two distinct aspects, corresponding to two etymologies for the suffix -nomial. According to some linguists, including Albert Dauzat, the "nomials" in polynomials and monomials come from the Greek onoma, meaning "name". This interpretation fits the algebraic perspective, in which the X in a polynomial is generally written as a capital letter and called the indeterminate or unknown. The use of the letter X dates back to René Descartes. But the idea of giving a problem's unknown a name is older still: it may go back to Diophantus (3rd century), who called it *arithmos*, "the number". Later, al-Khwarizmi (9th century) called it *shay, Arabic for "the thing". The practice reached France through the Spanish, who transcribed the word as xay. Descartes simplified it to its initial, giving us "X". Its use subsequently spread, particularly in legal contexts. In mathematics, the letter "X" is called an unknown when solving an equation, an indeterminate when dealing with polynomials, and a variable* in the context of functions. This triple role can sometimes be a source of confusion!
Albert Dauzat (1877–1955)
Another interpretation traces these "nomials" to the Greek nomos, meaning "law". This suggests a mathematical function, and hence a "rule" for computation. The interpretation fits the analytical perspective, in which *x* becomes a lowercase italic letter and is called the variable.
These two etymologies shed light on the subtle difference between a polynomial and a polynomial function. On the one hand, polynomials are formal objects; on the other, they are rules for computation. Thus, 2–3X2+4X4 is an object on which operations can be performed (addition, multiplication, division and so on), but it is also a function: the function that maps each value substituted for X (such as 5) to the number 2–3X2+4X4 (in this case, 2–3×52+4×54=2,427).
