Intereting Posts

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Are $\sigma$-algebras that aren't countably generated always sub-algebras of countably generated $\sigma$-algebras?
Motivation for triangle inequality
“Standard” ways of telling if an irreducible quartic polynomial has Galois group C_4?
Residue of complex function
The number of real roots of $1+x/1!+x^2/2!+x^3/3! + \cdots + x^6/6! =0$
Find the parametric form $S(u, v)$ where $a \le u \le b$ and $c \le v \le d$ for the triangle with vertices $(1, 1, 1), (4, 2, 1),$ and $(1, 2, 2)$.
Graham's Number : Why so big?
No. of possible solutions of given equation
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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?

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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.

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