What is the history behind the development of the term "coefficient"?

Question

Why are coefficients called "coefficients"?

For example I learned that squaring a number is called "squaring" because it actually refers to "making a square". That's how it was developed.

|<----+-----+---->|   3 

      squared


+-----+-----+-----+ 
|     |     |     |
|     |     |     |
|     |     |     |
+-----+-----+-----+ 
|     |     |     |
|     |     |     |
|     |     |     |
+-----+-----+-----+ 
|     |     |     |
|     |     |     |
|     |     |     |
+-----+-----+-----+ 

So what is efficient about a coefficient and why is it a "co" like a "coworker" or "coauthor" or a "coeditor"? I feel like if I understood it's history I might remember what it refers to.

Answer 1

com- from modern Latin meaning together.

efficient- also from modern Latin meaning accomplishing.

So it means to cooperate to produce a result.

Answer 2

The Oxford English Dictionary says:

According to Hutton, Vieta, who died in 1603, and wrote in Latin, introduced coefficiens in this sense.

and general meaning of the word around that time seems to have been

Cooperating to produce a result.

So perhaps the meaning is that the coefficient on $5 x$, namely the numeral $5$, cooperates with the value of $x$ to produce the result.