Polynomial

In algebra, a polynomial is, roughly speaking, a formal expression obtained from constants and one or several variables by making a finite number of additions, subtractions and multiplications. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 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.

Non-Examples

Expressions like ${\frac {x-1}{x^{2}+2}}$ or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}x^3+x-\sqrt{2}} , but this is only because Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}} and ${\sqrt {2}}$ are elements of the set (e.g. real numbers) that are being used as coefficients of the polynomials.

Expressions involving exponential functions, like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^x + 1} , are often mistaken for polynomials because of the similarity of notation. It is important to realize that the exponents appearing in a polynomial are always constants; when an expression has a variable in an exponent, such as the exponent x on the base 2 in the above expression, the expression is not a polynomial.

Monomials and simplification

The simplest polynomials are are the monomials having only one term. A monomial is either a constant, or a product of constants and positive whole number powers of variables. The constant is called the coefficient of the monomial. A coefficient equal to 1 is typically dropped from the notation, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} represents the same monomial as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1x^2} . Coefficients are usually written in as simplified a form as possible, but even then can sometimes look rather complicated, such as the coefficient in $(\pi +{\sqrt {3}}+1)x$. When a coefficient involves a sum, parentheses must be inserted around the coefficient.

Monomials are usually immediately simplified so that each variable is written once. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 y x} would be simplified by combing both powers of the variable x, producing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3 y} . When a monomial involves several variables, the order in which the variables are written is unimportant. By convention, the variables are typically listed in alphabetical order. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3 y} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y x^3} are considered to be the same monomial, but the latter would usually be rewritten as the former.

Polynomials are constructed by adding together a finite number of monomials. For instance, 1, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^4} , and $-3x^{2}yz^{3}$ are monomials, and their sum, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + 2x^4 - 3x^2yz^3} is a polynomial which is not a monomial.

There are two main procedures used in simplifying a polynomial. First, one can remove groupings by distributing, which produces a sum of monomials. For instance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x (x+2) + x(2x+2)} simplifies to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 4x + 2x^2 + 2x} . Second, one can combine like terms to reduce the number of monomials in the expression. For instance, the above polynomial can be further simplified to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^2 + 6x} by combining the two terms involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} and the two terms involving x.

Polynomials in one variable

In this section we deal with the simplest case, that is, polynomials involving only one variable, denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Degree and the Standard Form

After simplification, a polynomial can be written as a finite sum of monomials. It is sometimes useful to explicitly write a power of a variable in each monomial, even the constants. To accomplish this, you can write $x^{0}$ after the constant, so that 2 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^0} are considered the same.

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c x^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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + x} has degree 2, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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 simplifying as described above 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.