Polynomials are finite sums of power terms—no endless series
A polynomial combines variables and coefficients using only addition, subtraction, multiplication, and nonnegative integer powers, in finitely many terms. The word fuses Greek poly (many) with Latin nomen (name), extending binomial. From word problems to physics curves, polynomial equations and functions dominate modeling.
The symbol in a polynomial is called an indeterminate when the polynomial is treated as a formal expression, because it stands for no particular value, and a variable when the polynomial is read as a function; many writers use the two words interchangeably. Functional notation such as P(x) dates from a time when those ideas were not clearly separated, yet it remains handy for naming a polynomial and its indeterminate in one phrase. Substituting a number, another variable or any expression for x yields the associated polynomial function, and this works over any ring.
A single-indeterminate polynomial can always be arranged from a_n x^n down to a_0, with the a's as coefficients. Coefficients are usually numbers, though they may be any objects that can be added and multiplied and do not involve the indeterminates. A term's degree is the sum of its exponents, and the polynomial's degree is the largest degree among terms with nonzero coefficients, so 3x² − 5x + 4 has terms of degree two, one and zero.
Low degrees carry names: degree zero gives a constant, and degrees one, two and three give linear, quadratic and cubic polynomials. Names for higher degrees exist but see little use. The zero polynomial is the odd one out, since it has no terms at all; unlike other constants its degree is not zero but is either left undefined or set to a negative value such as −1.
Source: Polynomial
