A polynomial is a type of function that involves a finite sum of terms of the form anxn. Here, n is an integer greater than or equal to 0 and an can be any number, real or complex. The highest power of x is known as the "degree" and the an are called the coefficients. In general, the coefficients belong to what is known as a "ring".The word polynomial comes from replacing the Latin "bi" meaning two in "binomial" with the Greek "poly" meaning many. In general, a polynomial of degree n has the form:
where ai are the coefficients. An example of a polynomial is 3x2+5x-3 which is has degree 2 and is known as a quadratic. A polynomial equation is one where f(x) is set equal to 0.
Many problems in mathematics and science involve solving polynomial equations. For simple polynomials, there exist formulas for solving them such as the quadratic formula. For higher degrees, these formulas become incredibly complicated. However, no such formulas exist for an general nth degree polynomial so another method must be used. This is known as the Abel-Ruffini theorem.
The factor theorem states that for some polynomial f(x), if and only if f(a)=0 then f(x) has a factor (x-a). This means that f(x) can be written as (x-a) multiplied by another polynomial with degree n-1. This is most clearly seen through an example. Consider the cubic:
Substituting x=1 produces f(1)=0. Therefore, (x-1) is a factor and the polynomial can be written as:
The factor theorem allows us to simplyfy the polynomial we have to solve once we know one solution and this might make the problem easier to solve. There is also a similar theorem called the "remainder theorem". This states that if for some value of a, f(a) does not equal 0 then (x-a) cannot be a factor of f(x).
As with any equation, polynomials can be solved numerically using various methods, such as the Newton-Raphson method or the bisection method.
Polynomials in more than one variable
Polynomials can be extended to be functions of more than one variable such as x and y. They are defined in the same way so all variables have integer powers greater than equal to 0. However, there may be cross terms such as x2y3. In two variables, a generic polynomial can be expressed as:
A polynomial function is a function g(x) that is a polynomial in some other function f(x). For example, the following is a quadratic in f(x):
A power series is effectively a polynomial with an infinite number of terms. One example is Taylor series. A Taylor series can be used to describe any differentiable function. As power series contain an infinite number of terms, one must be careful as to whether they converge to a particular number or diverge to infinity.
Polynomials in computing
Some algorithms are said to perform in polynomial time. This means that the time for the algorithm to solve a problem is a polynomial function of the size of the problem. An example is the bubble sorting algorithm which can be used to sort a list of numbers. It has an efficiency of n2, so that doubling the size of the list increases the sorting time by 4. Different algorithms have different efficiencies, such as exponential and factorial.