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A '''polynomial''' in one [[variable]] <math>x</math> is a [[function]] <math>f(x)</math> of the form:
A '''polynomial''' is a type of [[function]] that involves a finite sum of terms of the form a<sub>n</sub>x<sup>n</sup>. Here, n is an [[integer]] greater than or equal to 0 and a<sub>n</sub> can be any number, [[real number|real]] or [[complex number|complex]].<ref>[https://www.merriam-webster.com/dictionary/polynomial merriam-webster.com]</ref> The highest power of x is known as the "degree" and the a<sub>n</sub> are called the [[coefficient]]s. In general, the coefficients belong to what is known as a "[[ring (mathematics)|ring]]".The word polynomial comes from replacing the Latin "bi" meaning two in "binomial" with the Greek "poly" meaning many.<ref>[http://www.etymonline.com/index.php?term=polynomial&allowed_in_frame=0 polynomial on etymonline.com]</ref> In general, a polynomial of degree n has the form:
<math>
<math>
f(x) = a_n x^n+a_{n-1}x^{n-1}+...+ a_2 x^2+a_1 x + a_0
f(x) = \sum^n_{i=0}{a_i x^i} = a_n x^n+a_{n-1}x^{n-1}+...+ a_2 x^2+a_1 x + a_0
</math>
</math>
where a<sub>i</sub> are the coefficients. An example of a polynomial is 3x<sup>2</sup>+5x-3 which is has degree 2 and is known as a [[quadratic equation|quadratic]]. A polynomial equation is one where f(x) is set equal to 0.
==Solving polynomials==
===Formulas===
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 n<sup>th</sup> degree polynomial so another method must be used.<ref>[http://mathworld.wolfram.com/AbelsImpossibilityTheorem.html Abel-Ruffini theorem on wolfram.com]</ref>
==Factor 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:
<math>
f(x) = x^3 - 2x + 6
</math>
Substituting x=1 produces f(1)=0. Therefore (x-1) is a factor and the polynomial can be written as:
<math>
<math>
\sum a_{ij}x^i y^j
f(x) = (x-1)(x^2-5x-6)
</math>,
</math>
The factor theorem allows us to simplyfy the polynomial we have to solve once we know one solution and this might make the porblem 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).
===Numerically===
As with any equation, polynomials can be solved numerically using various methods, such as the Newton-Raphson method or the bisection method.
==Generalizations==
===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 x<sup>2</sup>y<sup>3</sup>. In two variables, a generic polynomial can be expressed as:
<math>
f(x,y) = \sum_{i,j}{a_{ij} x^i y^j}
</math>
===Polynomial functions===
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 equation|quadratic]] in f(x):
<math>
g(x) = f(x)^2-7f(x)+2
</math>
A common type of polynomial functions are trigonometric polynomials where f(x) is replaced with [[trigonometry|trigonometric functions]] such as [[sine]]s or [[cosine]]s.
===Power Series===
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 [[differentiation|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 [[algorithm]]s 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 n<sup>2</sup>, so that doubling the size of the list increases the sorting time by 4.<ref>[http://www.geeksforgeeks.org/bubble-sort/ bubble sort algorithm on geeksforgeeks.org]</ref> Different algorithms have different efficiencies, such as [[exponential]] and [[factorial]].
==References==
{{reflist}}
==See also==
*[[Quadratic equation]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Algebra]]