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I was trying to figure out the distinction of a root and a zero and found people in such discussions make distinctions between equation vs. expression vs. polynomial without defining them. What is the difference exactly?

I'm especially confused by function. Likely I've forgotten many details from my high school math such as this definition. I can tell you about functions in several programming languages, but that's not the same thing at all.

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Here are informal definitions of the terms that seem confusing to you:

  • A function is a relation between two sets, usually sets of numbers. It maps elements of the first set to elements of the second set.
  • An expression is a combination of symbols representing a calculation, ultimately a number.
  • An equation describes that two expressions are identical (numerically).
  • A polynomial is a specific type of expression of one variable $x$, a sum of powers of $x$ multiplied by numbers.

I'm avoiding complexities and precision in the hope that this may start toward untangling.

A computer-language function takes input and produces output, which is analogous to a function mapping elements of the 1st set (input) to elements of the 2nd set (output).

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  • $\begingroup$ Thanks for your comments, but I guess I should have made it more clear that I was asking in the context of the zero vs. root thing. I believe that in reality all of these things are used as synonyms to equations, but specific types of equations, for example, a polynomial equation or an equation that expresses a function. For some reason people seem to think one term "zero" would be used with some equation types, while "root" would be used with others or even the term "solution". $\endgroup$
    – gnuarm
    Dec 25 '20 at 1:17
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    $\begingroup$ @gnuarm: The use of "root" and "zero" in this context varies, and although some people might insist on specific meanings for them, I'm fairly certain I could find many counterexamples to any such proposed usage throughout the literature (school level texts, college level texts, professional level monographs, etc.) if I had the desire to look (which I don't). In general, I've found that those who strongly insist on certain terms as having some specific mathematical meaning are often those who have not been very widely exposed to mathematical literature. $\endgroup$ Dec 25 '20 at 9:03
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    $\begingroup$ In my experience, the word “zero” is definitely a word that is only used for functions. Functions have zeros. “Roots” aren’t really used in a completely consistent way in my experience. $\endgroup$ Dec 25 '20 at 19:52
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    $\begingroup$ When I inherited my Algebra I classroom midyear, there was a poster on the wall that said "Expressions have roots, equations have solutions, functions have zeroes, graphs have x-intercepts." So that's out there, although my professional opinion is that roots and zeroes are synonymous and expressions actually have factors. Also, you need to be initiated into the guild of mathematicians before you get told what a polynomial really is. $\endgroup$ Dec 26 '20 at 2:59
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    $\begingroup$ @Matthew Daly: The poster usage is actually a nice summary of what (in retrospect) seems to be a reasonable format to follow, but as I'm sure you know, it's probably violated sufficiently that I wouldn't insist that students follow it. But it does ring rather "true" to me (e.g. polynomials have roots). Of course, I can imagine some smart-aleck student (which could have been me at times) asking the teacher about $x$-intercepts of distance-time graphs in physics or demand curves in economics, where there is no $x.$ $\endgroup$ Dec 26 '20 at 16:34

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