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This may seem like a very basic question, but:

What exactly is the difference between a root of a polynomial, and a zero? Of course I realise that they are technically exactly the same thing, but there seem to be subtle rules as to when to use each term, and a couple of times in the past I have been told I am using "root" where I should be using "zero".

Is it generally accepted that one should use "root" in an algebraic context, and "zero" in a analytic context? If not, when should one use one or the other...and does it really matter?!

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up vote 12 down vote accepted

The difference is the following: whenever you have a function $f:X\to \mathbb R$ you say that $x^*$ is a zero of $f$ if $f(x^*) = 0$. On the other hand, the same $x^*$ is a root of the equation $f(x) = 0$. The rule of thumbs: zero refers to the function (e.g. polynomial) and root refers to the equation.

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I've often heard the two terms used interchangeably, though. "Roots of the polynomial $x^3+x-1$" isn't something I'd be surprised to hear a professional mathematician say. – user7530 Nov 16 '11 at 11:05

In general we will find zero of a function root of an equation

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Welcome to MSE! I realize you don't yet have enough reputation, but this would have been better as a comment. Regards – Amzoti May 18 '13 at 17:51

Don't use "root" in Australia. It has a very different meaning.

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Wouldn't this answer be better as a comment? – lhf Nov 16 '11 at 12:11
@lhf, you have a point. I figure the question was, in part, "when should you use 'root'" and I figure that "not in Australia" is a (partial) answer to that question. I don't see where any harm is done by posting it as an answer rather than as a comment. And there is the advantage that you can't vote down a comment. – Gerry Myerson Nov 16 '11 at 12:19
Thanks Gerry. So in Australia, the root of a polynomial is considered to be a deviant act, as opposed to a mathematical object? I'll keep this in mind when I'm next there. – mrb Mar 9 '12 at 10:51

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