# Difference between root, zero and solution.

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?

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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.

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.