# Polynomials: A Brief Description

In order to understand mathematics, first, you have to speak the language. The problem is, with mathematics, unlike French or German or any other spoken language, this dialect is replete with difficulty and at times quite incomprehensible. And in mathematics, even when you understand something, you still often feel as though you really need to understand it even more. Thus, it certainly helps if you can get off to a good start by at least understanding some of the languages. Here we give insight into what a polynomial is and then we will continue their operations like multiplying polynomials, adding, subtracting them.

A polynomial is an expression found in algebra. Technically a polynomial of degree n is an expression of the form a(n)x^n + a(n-1)x^(n-1) +… + a(1)x + a(0), where each of the a(n) terms corresponds to some integer, the n-terms following the x^ correspond to the exponents and are positive integers, and n and a(n) are not equal to 0 (if they were then this would not be a polynomial of degree n). In plain English, a polynomial is any expression such as 3x^4 + 2x^3 – x + 4, or 2x^2 – 3x + 1. The degree is the highest exponent that occurs in the expression. Thus, the first polynomial is of degree 4 and the second is of degree 2.

The first few polynomials, those of degree 1, 2, 3, 4 and 5 have special names. A first-degree polynomial is a linear function because its graph produces a line. The second, a quadratic; the third a cubic; the fourth a quartic; and the fifth, a quantic. After these, the polynomial is generally referred to by its degree.

The above-written polynomial uses the variable x, and this is most common; however, we could just as easily have written a polynomial in some other letter or variable, and some common other choices would be the letters y or t. Bear in mind that changing the letter in which the polynomial is written does not alter the nature or behaviour in any way.

Polynomials are just one kind of algebraic expression. They are very useful in modelling many real-world problems, and they occur in many formulas. In more advanced courses, polynomials are encountered to serve as substitutes for other functions for which no apparent similarity is evident. Thus, the amazing versatility of polynomials. Multiplying polynomials is very easy, and you should be knowing how to do the same. Otherwise, you will not be able to solve any algebraic problems.

As far as the pictures, or graphs, of polynomial functions, they look somewhat like roller coasters, often with many hills and valleys. These curves are “smooth” in the sense that they have no sharp turns or corners and can be drawn all in one piece. For this reason, these polynomial functions play an important role in the branch of mathematics called analysis and serve as an important tool in many other branches as well. So, do learn the world of polynomial and conquer the mathematics.