Definition of root of equation/expression

Question

Related: Where does the word “roots” come from when talking about zeros

A student recently wrote:

The positive root of 3 sin x = x is near 2.

I am questioning the student's use of of the word root.

Stewart-Redlin-Watson do not define the root of an equation or expression at all. However, in the section titled "Real Zeros of Polynomials," after spending most of the section referring to the zeros of polynomials, an exercise appears stating:

Show that the equation (fifth degree polynomial) = 0 has exactly one rational root...

They are careful with the definition of a zero:

If P is a polynomial function, then c is called a zero of P if P(c)=0. In other words, the zeros of P are the solutions of the polynomial equation P(c)=0.

Sullivan's Precalculus has a very similar approach, and further the term root is essentially defined as a synonym to the term zero.

So polynomial functions have zeros, or roots, so we could also say polynomial expressions have zeros or roots if we say "well just regard that polynomial as a function." But do equations have zeros or roots? It seems that equations just have solutions, and graphs have x intercepts.

Followup question: is it just polynomials or algebraic expressions that have roots? Certainly, the history of the term comes from the case of looking for nth roots of numbers. Do transcendental functions have zeros, roots, both, or neither?

Answer 1

Equations have solutions, not zeros or roots. A solution to an equation is an assignment of values to each indeterminate in the equation that makes the equation true.

Functions have zeros, also called roots. A zero of a function is an element of the domain of the function such that the corresponding value of the function is zero. "Roots" is most commonly used for polynomial functions, but I don't see any reason not to talk about roots of functions more generally, as a synonym of "zeros". (Similarly, polynomial expressions have roots/zeros, by looking at the corresponding polynomial function.)

I would correct a student on using "zero" or "root" for a solution to an equation. I think it's perfectly fine to use "zero" and "root" interchangeably for functions, whether they're polynomial or not. In fact, I'd explicitly define them as synonyms from the outset; I can't think of a situation where it's useful to distinguish between the two terms.

Answer 2

Do transcendental functions have zeros, roots, both, or neither?

The expression "roots" is also used for transcendental functions, and means the same thing as "zeros." Namely, a root of a function $f$ is an $x_0$ (in an explicitly or implicitly specified domain) such that $f(x_0)=0$.

Or, like you quoted, one might talk about the roots of an equation $f(x)= 0$ to refer to the set of its solutions. This is possibly a bit old-fashioned, but still not uncommon.

Some authors even extend the usage to expressions of the form $f(x)=g(x)$, that is equations. At least in the case where the right-hand side is a constant or the situation is otherwise concrete.

Thus to answer your question:

But do equations have zeros or roots?

Some authors refer to solutions of an equation as roots of an equation. It is possibly a bit old-fashioned and usually confined to certain contexts, which would however encompass the particular one your mentioned.

However, to talk about the zeros of an equation in general would not make sense.

Note that historically it is not the case that this usage of "root" was extrapolated from the usage in square-root and alike. It is rather the other way around. It was generically used somewhat like an unknown and one might rather think of $7$-th root of $2$, as short for a root of the equation $x^7=2$.

Arguably, by now this general usage of root is possibly outdated and could cause confusion. I do not advocate for it, but would stop short of calling it wrong.