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We often use the word roots when referring to the solutions of an equation. For instance, when we have a polynomial $P(x)$, we call its zeros the roots of $P(x)$.

For some polynomials we can relate the zeros with a root function of some kind, say $$ x^2-4=0, $$ we can take the square roots of $4$ to get the solutions. However with some functions, like trigonometrics, there doesn't seem to be a clear relation with the typical root functions.

I ask because in an assignment I gave, students where asked to find the roots of a rational function. I found out that many students had never heard the word roots used in a "find the zeros" context. Many of them took the square roots of the function and tried solving some equations that made no sense.

When looking online, I could not find any solid reference to the first uses of the word in that context.

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  • $\begingroup$ I was always taught that functions have zeros, equations have solutions, graphs have x-intercepts and polynomials (as in their own right distinct from the concept of function) have roots. $\endgroup$ Sep 30, 2014 at 19:51
  • $\begingroup$ @DavidButlerUofA Granted, $ab-c$ is a polynomial independently of any consideration of which symbols represent unknowns. But in order to state what roots it has, you need to give formula(s) for its unknown(s), which entails stating which symbols represent unknowns. To state that the root of $ax-c$ is $x=c/a$, you need to establish that $x$ represents an unknown, and your polynomial now represents a function of $x$. $\endgroup$
    – Rosie F
    Oct 21, 2022 at 7:49

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Doctor Peterson in The Math Forum refers to the following sources.

From D. E. Smith, History Of Mathematics Vol II (1925), footnote page 393:

The Arabs also used jidr (dyizr, root), whence the Latin radix.

A longer quote from W. W. R. Ball, A Short Account of the History of Mathematics (4th edition, 1908):

The algebra of Alkarismi holds a most important place in the history of mathematics, for we may say that the subsequent Arab and the early medieval works on algebra were founded on it, [...]. The unknown quantity is termed either "the thing" or "the root" (that is, of a plant), and from the latter phrase our use of the word root as applied to the solution of an equation is derived. The square of the unknown is called "the power".

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  • $\begingroup$ It was a simple Google search :) Both books are available online. $\endgroup$ Sep 30, 2014 at 19:19
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    $\begingroup$ I had found a similar page by Dr Peteron, but it was not as precise. I tossed other results in google because of that. Shame on me, woot for you! $\endgroup$ Sep 30, 2014 at 19:51
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    $\begingroup$ @Jean-Sébastien I meant finding the books with the citations was a simple Google search $\endgroup$ Sep 30, 2014 at 19:55

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