There is this question that asks to find the first 3 terms in the binomial expansion of $(x+\frac{1}{x^2})^6$. The question itself is easy, but someone asks me $x+\frac{1}{x^2}$ is not a polynomial by definition, but how come all textbooks use the word "find its binomial expansion"? How come in maths we use some defined terms wrongly?
How can I explain this?
The easiest escape is: as long as there are 2 terms, we consider it a binomial. I hope there are some better explanations than this.
Helps are greatly appreciated.
The term "binomial" is much older than "polynomial". The first dates back (at least) to Fibonacci in 1202, the other to Viete in 1591 (ed: after Stevins introduced "multinomial" in 1585, before that "universal (sum)" was used). In general, a binomial is just a group, usually a sum, of two terms. Thus the binomial theorems allow the expansion of powers of such sums. Which was extensively used to approximate roots, even long before this method was systematized as Newton's method by Simpson some years after 1700.
Sources
Note also that in biology, an name of a species with two parts is a "binomial" or "binominal", and a name with three parts a "trinomial" or "trinominal".