# Difference between revisions of "Polynomial"

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− | A polynomial is a function in one or more | + | A '''polynomial''' is a [[function]] in one or more [[variable]]s that consists of a sum of variables raised to [[nonnegative]], [[integer|integral]] powers and multiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[real]] or [[complex]] numbers; but in [[abstract algebra]] often an arbitrary [[field]]). |

For example, these are polynomials: | For example, these are polynomials: | ||

− | * <math>4x^2 + 6x - 9</math>, in the variable x | + | * <math>4x^2 + 6x - 9</math>, in the variable <math>x</math> |

− | * <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables x and y | + | * <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math> |

− | * <math>5x^4 - 2x^2 + 9</math>, in the variable x | + | * <math>5x^4 - 2x^2 + 9</math>, in the variable <math>x</math> |

* <math>\sin^2{x} + 5</math>, in the variable <math>\sin x</math> | * <math>\sin^2{x} + 5</math>, in the variable <math>\sin x</math> | ||

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* <math>\sin^2{x} + 5</math> | * <math>\sin^2{x} + 5</math> | ||

* <math>\frac{4x+3}{2x-9}</math> | * <math>\frac{4x+3}{2x-9}</math> | ||

− | are functions, but ''not'' polynomials, in the variable x | + | *<math>x^{-1}+2+3x+x^2</math> |

+ | *<math>x^{1/3}=\sqrt[3]{x}</math> | ||

+ | are functions, but ''not'' polynomials, in the variable <math>x</math> | ||

==Introductory Topics== | ==Introductory Topics== | ||

+ | ===A More Precise Definition=== | ||

+ | |||

+ | A polynomial in one variable is a function <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. Here, <math>a_i</math> is the <math>i</math>th coefficient and <math>a_n \neq 0</math>. Often, the leading coefficient of a polynomial will be equal to 1. In this case, we say we have a ''monic'' polynomial. | ||

+ | |||

+ | === The Degree of a Polynomial === | ||

− | === | + | The simplest piece of information that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as the ''degree'' of the polynomial and is written <math>\deg(P)</math>. For instance, <math>\deg(x^2 + 3x + 4) = 2</math> and <math>\deg(x^5 - 1) = 5</math>. When a polynomial is written in the form <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math> with <math>a_n \neq 0</math>, the integer <math>n</math> is the degree of the polynomial. |

− | + | The degree, together with the coefficient of the largest term, provides a surprisingly large amount of information about the polynomial: how it behaves in the [[limit]] as the variable grows very large (either in the positive or negative direction) and how many roots it has. | |

===Finding Roots of Polynomials=== | ===Finding Roots of Polynomials=== | ||

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====What is a root?==== | ====What is a root?==== | ||

− | A root is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>. For some polynomials, you can easily set the polynomial equal to zero and solve | + | A [[root]] is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of <math>x^2 - 4</math> because <math>2^2 - 4 = 0</math>. For some polynomials, you can easily set the polynomial equal to zero and solve or otherwise find roots, but in some cases it is much more complicated. |

====The Fundamental Theorem of Algebra==== | ====The Fundamental Theorem of Algebra==== | ||

− | The [[ | + | The [[Fundamental Theorem of Algebra]] states that any polynomial with [[complex number|complex]] coefficients can be written as |

− | <math>P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)</math> where <math>k</math> is a constant, the <math>x_i</math> are (not necessarily distinct) complex numbers and <math>n</math> is the | + | <math>P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)</math> where <math>k</math> is a constant, the <math>x_i</math> are (not necessarily distinct) complex numbers and <math>n</math> is the degree of the polynomial in exactly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots will be <math>x_1,x_2,...,x_n</math>. This also tells us that the degree of a given polynomial is at least as large as the number of distinct roots of that polynomial. In quadratics roots are more complex and can simply be the square root of a prime number. |

====Factoring==== | ====Factoring==== | ||

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====The Rational Root Theorem==== | ====The Rational Root Theorem==== | ||

− | We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. | + | We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>. The [[Rational Root Theorem]] states that if <math>P(x)</math> has a rational root <math>\pm\frac{p}{q}</math> and this [[fraction]] is fully reduced, then <math>p</math> is a [[divisor]] of <math>a_0</math> and <math>q</math> is a divisor of <math>a_n</math>. This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It is also especially convenient when dealing with monic polynomials. |

− | ==== | + | ====Descarte's Law of Signs==== |

− | By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not | + | By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decarte's Rule of Signs says that for a polynomial <math>P(x)</math>, the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of <math>P(-x)</math>, or is less than that by a multiple of 2. |

− | + | ===Binomial Theorem=== | |

− | [[Binomial | + | The [[Binomial Theorem]] can be very useful for factoring and expanding polynomials. |

+ | |||

+ | |||

+ | ===Special Values=== | ||

+ | Given the coefficients of a polynomial, it is very easy to figure out the value of the polynomial on different inputs. In some cases, the reverse is also true. The most obvious example is also the simplest: for any polynomial <math>P(x) = a_nx^n + \ldots + a_1 x + a_0</math>, <math>P(0) = a_0</math> so the value of a polynomial at 0 is also the constant coefficient. | ||

+ | |||

+ | Similarly, <math>P(1) = a_n + a_{n - 1} + \ldots + a_1 + a_0</math>, so the value at 1 is equal to the sum of the coefficients. | ||

+ | |||

+ | In fact, the value at any point gives us a linear equation in the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given the value of a polynomial <math>P</math> and <math>\deg(P) + 1</math> different points, we can always find the coefficients of the polynomial. | ||

==Intermediate Topics== | ==Intermediate Topics== | ||

+ | *[[Complex numbers]] | ||

+ | *[[Fundamental Theorem of Algebra]] | ||

+ | *[[Roots of unity]] | ||

==Olympiad Topics== | ==Olympiad Topics== | ||

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* [[Newton's identities]] | * [[Newton's identities]] | ||

* [[Newton sums]] | * [[Newton sums]] | ||

− | |||

− | |||

− | |||

− | |||

== See also == | == See also == | ||

* [[Algebra]] | * [[Algebra]] | ||

+ | |||

+ | [[Category:Algebra]] | ||

+ | [[Category:Polynomials]] | ||

+ | [[Category:Definition]] |

## Latest revision as of 03:50, 31 March 2023

A **polynomial** is a function in one or more variables that consists of a sum of variables raised to nonnegative, integral powers and multiplied by coefficients from a predetermined set (usually the set of integers; rational, real or complex numbers; but in abstract algebra often an arbitrary field).

For example, these are polynomials:

- , in the variable
- , in the variables and
- , in the variable
- , in the variable

However,

are functions, but *not* polynomials, in the variable

## Introductory Topics

### A More Precise Definition

A polynomial in one variable is a function . Here, is the th coefficient and . Often, the leading coefficient of a polynomial will be equal to 1. In this case, we say we have a *monic* polynomial.

### The Degree of a Polynomial

The simplest piece of information that one can have about a polynomial of one variable is the highest power of the variable which appears in the polynomial. This number is known as the *degree* of the polynomial and is written . For instance, and . When a polynomial is written in the form with , the integer is the degree of the polynomial.

The degree, together with the coefficient of the largest term, provides a surprisingly large amount of information about the polynomial: how it behaves in the limit as the variable grows very large (either in the positive or negative direction) and how many roots it has.

### Finding Roots of Polynomials

#### What is a root?

A root is a value for a variable that will make the polynomial equal zero. For an example, 2 is a root of because . For some polynomials, you can easily set the polynomial equal to zero and solve or otherwise find roots, but in some cases it is much more complicated.

#### The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that any polynomial with complex coefficients can be written as

where is a constant, the are (not necessarily distinct) complex numbers and is the degree of the polynomial in exactly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots will be . This also tells us that the degree of a given polynomial is at least as large as the number of distinct roots of that polynomial. In quadratics roots are more complex and can simply be the square root of a prime number.

#### Factoring

Different methods of factoring can help find roots of polynomials. Consider this polynomial:

This polynomial easily factors to:

Now, the roots of the polynomial are clearly -3, -2, and 2.

#### The Rational Root Theorem

We are often interested in finding the roots of polynomials with integral coefficients. Consider such a polynomial . The Rational Root Theorem states that if has a rational root and this fraction is fully reduced, then is a divisor of and is a divisor of . This is convenient because it means we must check only a small number of cases to find all rational roots of many polynomials. It is also especially convenient when dealing with monic polynomials.

#### Descarte's Law of Signs

By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not necessarily real) of a polynomial of degree n is n. This tells us nothing about whether or not these roots are positive or negative. Decarte's Rule of Signs says that for a polynomial , the number of positive roots to the equation is equal to the number of sign changes in the coefficients of the polynomial, or is less than that number by a multiple of 2. The number of negative roots to the equation is the number of sign changes in the coefficients of , or is less than that by a multiple of 2.

### Binomial Theorem

The Binomial Theorem can be very useful for factoring and expanding polynomials.

### Special Values

Given the coefficients of a polynomial, it is very easy to figure out the value of the polynomial on different inputs. In some cases, the reverse is also true. The most obvious example is also the simplest: for any polynomial , so the value of a polynomial at 0 is also the constant coefficient.

Similarly, , so the value at 1 is equal to the sum of the coefficients.

In fact, the value at any point gives us a linear equation in the coefficients of the polynomial. We can solve this system and find a unique solution when we have as many equations as we do coefficients. Thus, given the value of a polynomial and different points, we can always find the coefficients of the polynomial.