Difference between revisions of "Polynomial"

Revision as of 14:08, August 24, 2017

A polynomial is a type of function that involves a finite sum of terms of the form a_nxⁿ. Here, n is an integer greater than or equal to 0 and a_n can be any number, real or complex.^[1] The highest power of x is known as the "degree" and the a_n 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.^[2] In general, a polynomial of degree n has the form:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 }

where a_i 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.

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^th degree polynomial so another method must be used.^[3]

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^3 - 2x + 6 }

Substituting x=1 produces f(1)=0. Therefore (x-1) is a factor and the polynomial can be written as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = (x-1)(x^2-5x-6) }

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 x²y³. In two variables, a generic polynomial can be expressed as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y) = \sum_{i,j}{a_{ij} x^i y^j} }

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 in f(x):

$g(x)=f(x)^{2}-7f(x)+2$

A common type of polynomial functions are trigonometric polynomials where f(x) is replaced with trigonometric functions such as sines or cosines.

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 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.^[4] Different algorithms have different efficiencies, such as exponential and factorial.

@@ Line 1: / Line 1: @@
-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>
-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>
-In elementary mathematics, the [[coefficient]]s <math>a_i</math> are typically chosen to be [[real]] or [[complex number]]s. However, it makes sense to define a polynomial with coefficients in any [[ring]].
+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.
-The largest power of <math>x</math> that appears in the polynomial is called the [[degree]] of the polynomial.
+==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>
-*Example: <math>f(x) = 4x^3-3x+1</math> is a degree 3 polynomial with integer coefficients.
+==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:
-A polynomial in two variables <math>x,y</math> is, similarly, a finite sum
+<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>
-\sum a_{ij}x^i y^j
+f(x) = (x-1)(x^2-5x-6)
-</math>,
+</math>
-where the coefficients <math>a_{ij}</math> are elements of some [[ring]]. Polynomials in 3 or more variables are defined similarly.
+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]].
-When we substitute a value ''b'' into the polynomial ''f(x)'' then we get an actual number as output. This number is the value <math>a_nb^n +a_{n-1}b^{n-1}+...+ a_2 b^2+a_1b +a_0</math>. This process is called "evaluating the polynomial at ''b''".
+==References==
+{{reflist}}
-Some [[algorithm]]s are said to perform in polynomial time. There are algorithms which can factor polynomials in an amount of time that is a polynomial n<sup>k</sup>.
+==See also==
+*[[Quadratic equation]]
+[[Category:Mathematics]]
 [[Category:Algebra]]

Difference between revisions of "Polynomial"

Revision as of 14:08, August 24, 2017

Contents

Solving polynomials

Formulas

Factor Theorem

Numerically

Generalizations

Polynomials in more than one variable

Polynomial functions

Power Series

Polynomials in computing

References

See also

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