A polynomial simply means many (poly-) terms (nom-) and is just a made-up word for a description of a sum of terms that are individually expressed as a variable (unknown) usually with a positive whole number exponent and a multiplying constant. So when such terms are strung together in a sum, mathematicians called the sum a polynomial.
The simplest polynomial is ax+b because we have an unknown x and constants (numbers) a and b. Such a simple algebraic sum isn't usually called a polynomial because the word is normally reserved for greater powers or exponents of the unknown variable.
So the word polynomial was invented as a name to cover all such algebraic sums. Just as the word elephant is used to describe the animal without going into a long description involving size, colour and long nose or trunk. The elephant existed before anyone invented a word for it. The same is true for polynomials.
As for when polynomials (as we call them) were first introduced as a mathematical concept, well the date is probably around 300BC, but at that time the word polynomial hadn't been invented even though a descriptive word could have been used. But it seems the ancient Egyptians and Babylonians were solving linear and quadratic equations (which are polynomials) without "knowing" they were working with polynomials. They just called them something else in their own language. Polynomials date back to a time not long after algebra was introduced as a technique for representing unknown quantities. The word algebra is likely to have been derived from the Arabic word al-jabru, meaning restoration and balancing, the process of solving an equation.