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?