Polynomial

In algebra, a polynomial is, roughly speaking, a formal expression obtained from constant numbers called coefficients and one or several unspecified numbers called variables by making a finite number of additions, subtractions and multiplications. For instance, $x^{2}-2x+1$ is a polynomial involving one variable, x (often called a polynomial in one variable), whereas $x^{2}+y^{2}$ is a polynomial in two variables, $x$ and $y$.

Polynomials are an essential element of most applications of mathematics to a systematic study of physical problems and phenomena. Polynomials are the most basic objects that can be used to represent situations where a certain quantity is unknown, like the time it will take for a ball thrown in the air to fall back to the ground. They are also the most basic objects that can be used to represent a quantity that varies. For instance, the rate at which a chemical reaction proceeds often varies as the chemistry|concentrations of the reactants change. This variation is often specified by a polynomial whose variables are the concentrations.

The study of many phenomena requires more sophisticated objects than polynomials. Sometimes such analysis is so complicated that it becomes essential to approximate certain objects with polynomials. Approximation of sophisticated mathematical objects with polynomials is an important application of calculus, and is an essential tool in numerical analysis.

The polynomials encountered first in school and occurring most commonly in applications have real number coefficients. However, there are applications where other types of coefficients occur. For instance, the study of very small phenomena using quantum mechanics requires the use of complex numbers. Error-correcting codes, used to avoid loss or corruption of data, involve coefficients governed by arithmetic modulo 2. Polynomials with complex number coefficients or coefficients modulo 2 behave similarly to polynomials with real number coefficients in many ways. The similarities and differences are discussed on the advanced version of this page. Polynomials with even more general types of coefficients are important in advanced mathematics, and are discussed on the polynomial ring page. For the rest of the present article, all polynomials considered will have real number coefficients.

Polynomials in one variable

In this section we deal with the simplest case, that is, polynomials involving only one variable, denoted $x$.

Examples and Non-examples

After simplification, a polynomial can be written as a finite sum of terms, called monomials. Each monomial is either a constant, or a constant times a positive whole number power of x. For instance, 1, $2x^{4}$, and $-3x^{2}$ are monomials, and their sum, $2x^{4}-3x^{2}+1$ is a polynomial.

Expressions like ${\frac {x-1}{x^{2}+2}}$ or ${\sqrt {x^{2}+1}}$ are not polynomials ; the first one is a rational function, and the second one is an irrational expression, due to the square root symbol. Such operations might be expressed within the constant numbers, as in the example ${\frac {1}{2}}x^{3}+x-{\sqrt {2}}$, but this is only because ${\frac {1}{2}}$ and ${\sqrt {2}}$ are elements of the set (e.g. real numbers) that are being used as coefficients of the polynomials.

A coefficient equal to 1 in front of a positive power of x is typically dropped from the notation, so that $x^{2}-x+1$ represents the same polynomial as $1x^{2}-1x+1$. It is sometimes useful to explicitly write a power of x in each monomial, even the constants. To accomplish this, you can write $x^{0}$ after the constant, so that 2 and $2x^{0}$ are considered the same.

Degree and the Standard Form

The power of the variable appearing in a monomial is the degree of the monomial. By the above convention, a constant c is the same as $cx^{0}$ and has degree equal to 0. The degree of a polynomial is the largest of the degrees of the monomials appearing in the polynomial. The only exception is the constant polynomial 0, which typically is not assigned a degree (for reasons made clear below). As an example, 2 has degree 0, $x^{2}+x$ has degree 2, and $-3x^{4}+x^{2}+x^{5}$ has degree 5.

The degree is an important identifier when working with polynomials. For instance, many procedures for factoring or solving polynomial equations require identifying the degree of the polynomial first. In the last example above, we had to scan through the polynomial from the left all the way through the right to determine that the degree is 5. To facilitate identifying the degree of a polynomial, as well as manipulations of polynomials, they are usually written in standard form. The standard form of a polynomial is obtained by combining terms of the same degree, and then writing the monomials so that the exponents decrease from left to right. The degree of a polynomial in standard form is the degree of the first monomial appearing. The term of highest degree is the leading term and its coefficient is the leading coefficient. A polynomial with leading coefficient equal to 1 is monic. We can put the last example above in standard form by rearranging the monomials to obtain $x^{5}-3x^{4}+x^{2}$. It is of course just as easy to work with polynomials where the monomials are written so that the degrees increase from left to right.