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Can somebody precisely tell me what is the difference between a root, a zero and solution ?

Is it correct to say that an equation has solutions, and a polynomial has zeros or roots?

Cameron Buie
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2 Answers2

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Here is my interpretation.

The word solution is used in the following context. Find the solution to $$f(x) = b \tag{$\star$},$$ where $f: A \mapsto B$, i.e., find the set of all $x \in A$ such that $f(x) = b$.

The zeros of the function $f$ is the set of all $a \in A$ such that $f(a) = 0_B$, where $0_B$ is the zero element in $B$. Hence, zeros are used specifically in the context when the $b$ in $(\star)$ is $0_B$. To see it in a slightly different way, the zeros of $g(x)$, where $$g(x) = f(x)-b$$ are the solutions to the equation $f(x) = b$.

The term roots are typically used to describe the zeros of a function, when the function $f(x)$ is of the following form: $f: R \mapsto R$, where $R$ is a ring. I believe, the usage was due to the fact that we talk about finding the square roots, cube roots, etc, which was then extended to polynomials and thereby extended to rings.

  • (+1): I think you mean "$a\in A$ such that $f(a)=0_B$", perhaps. – Cameron Buie Apr 23 '13 at 21:48
  • @CameronBuie Thanks. Updated. –  Apr 23 '13 at 21:49
  • The term 'root' is often used to describe a solution to an equation. Consider the Roots function in Mathematica. In contrast, Maple's Roots function matches the definition here. It is unclear to me that there is a canonical definition. If so, I'd like that to be documented. – Alan Apr 15 '16 at 17:43
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If we're talking about an equation, a root and a solution are synonymous.

If we're talking about a function, a root and a zero are synonymous.

Cameron Buie
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