For example:
is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree is 2 + 1 = 3.
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:
It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, variable x, and exponent 2. In the second term, the coefficient is –5. The third term is a constant. Since the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.
Two terms with the same variables raised to the same powers are called "like terms", and they can be combined (after having been made adjacent) using the distributive law into a single term, whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0, in which case their combination just cancels out the terms. Polynomials can be added using the associative law of addition (which simply groups all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if
then
which can be simplified to
To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if
then
which can be simplified to
The sum or product of two polynomials is always a polynomial.
Alternative forms
In general any expression can be considered to be a polynomial if it is built up from variables and constants using only addition, subtraction, multiplication, and raising expressions to constant positive whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)3 is a polynomial; its standard form is x3 + 3x2 + 3x + 1.
Division of one polynomial by another does not, in general, produce a polynomial, but rather produces a quotient and a remainder.[6] A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,
is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to and is just a constant. When this expression is used as a term, its coefficient is therefore . For similar reasons, if complex coefficients are allowed, one may have a single term like ; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.
is not a polynomial because it includes division by a non-constant polynomial.
is not a polynomial, because it contains a variable used as exponent.
Since subtraction can be replaced by addition of the opposite quantity, and since positive whole number exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and variables using only addition and multiplication.
Polynomial functions
A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies
for all arguments x, where n is a non-negative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by
is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in
An example is also the function which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that (see Chebyshev polynomials).
Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.
Polynomial equations
Main article: Algebraic equation
A polynomial equation, also called algebraic equation, is an equation in which a polynomial is set equal to another polynomial.
is a polynomial equation. In case of a univariate polynomial equation, the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x2 – y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.
