Difference between revisions of "Polynomial"

Latest revision as of 14:57, 30 August 2024

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). Note that a constant is also a polynomial.

For example, these are polynomials:

$4x^2 + 6x - 9$ , in the variable $x$
$x^3 + 3x^2y + 3xy^2 + y^3$ , in the variables $x$ and $y$
$5x^4 - 2x^2 + 9$ , in the variable $x$
$\sin^2{x} + 5$ , in the variable $\sin x$
$35$ , in any variable
$2^x+2.34(2^x)^3-\sqrt2(2^x)^2$ , in the variable $2^x$
$(3+i)y^{100}$ , in the variable $y$

However,

$\sin^2{x} + 5$
$\frac{4x+3}{2x-9}$
$x^{-1}+2+3x+x^2$
$x^{1/3}=\sqrt[3]{x}$
$3x^2-4x+5+cos(x)$
$x^2+2^x$
$3\sqrt{x}+9x-17x^2\sqrt2$

are functions, but not polynomials, in the variable $x$

Introductory Topics

A More Precise Definition

A polynomial in one variable is a function $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ . Here, $a_i$ is the $i$ th coefficient and $a_n \neq 0$ . 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 $\deg(P)$ . For instance, $\deg(x^2 + 3x + 4) = 2$ and $\deg(x^5 - 1) = 5$ . When a polynomial is written in the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ with $a_n \neq 0$ , the integer $n$ 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 $x^2 - 4$ because $2^2 - 4 = 0$ . 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

$P(x) = k(x-x_1)(x-x_2)\cdots(x-x_n)$ where $k$ is a constant, the $x_i$ are (not necessarily distinct) complex numbers and $n$ 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 $x_1,x_2,...,x_n$ . 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:

$x^3 + 3x^2 - 4x - 12 = 0$

This polynomial easily factors to:

$(x+3)(x^2-4) = 0$

$(x+3)(x-2)(x+2) = 0$

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 $P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0$ . The Rational Root Theorem states that if $P(x)$ has a rational root $\pm\frac{p}{q}$ and this fraction is fully reduced, then $p$ is a divisor of $a_0$ and $q$ is a divisor of $a_n$ . 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. Descarte's Rule of Signs says that for a polynomial $P(x)$ , 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 $P(-x)$ , 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 $P(x) = a_nx^n + \ldots + a_1 x + a_0$ , $P(0) = a_0$ so the value of a polynomial at 0 is also the constant coefficient.

Similarly, $P(1) = a_n + a_{n - 1} + \ldots + a_1 + a_0$ , 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 $P$ and $\deg(P) + 1$ different points, we can always find the coefficients of the polynomial.

@@ Line 1: / Line 1: @@
-A polynomial is a function in one or more variables that consists of a sum of variables raised to [[integer|integral]] powers and multiplied by coefficients.
+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]]). Note that a [[constant]] is also a polynomial.
 For example, these are polynomials:
-* <math>4x^2 + 6x - 9</math>
+* <math>4x^2 + 6x - 9</math>, in the variable <math>x</math>
-* <math>x^3 + 3x^2y + 3xy^2 + y^3</math>
+* <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>
+* <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>35</math>, in any variable
+* <math>2^x+2.34(2^x)^3-\sqrt2(2^x)^2</math>, in the variable <math>2^x</math>
+* <math>(3+i)y^{100}</math>, in the variable <math>y</math>
-These '''aren't''' polynomials:
+However,
 * <math>\sin^2{x} + 5</math>
 * <math>\frac{4x+3}{2x-9}</math>
+* <math>x^{-1}+2+3x+x^2</math>
+* <math>x^{1/3}=\sqrt[3]{x}</math>
+* <math>3x^2-4x+5+cos(x)</math>
+* <math>x^2+2^x</math>
+* <math>3\sqrt{x}+9x-17x^2\sqrt2</math>
+are functions, but ''not'' polynomials, in the variable <math>x</math>
+==Introductory Topics==
+===A More Precise Definition===
-==Introductory Topics==
+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 ===
-===A More Precise Definition===
+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.
-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_n</math> is the <math>n</math>th coefficient, and <math>n</math> is an integer.
+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===
@@ Line 21: / Line 35: @@
 ====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 the equations to find roots, but in some cases it is much more complicated.
+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]] states that any polynomial with [[complex number|complex]] coefficients can be written as
+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 highest power of <math>x</math> that <math>P(x)</math> contains (also called the ''degree'').  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.
+<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====
@@ Line 44: / Line 58: @@
 ====The Rational Root Theorem====
-For a polynomial <math>P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0</math>, let p be a factor of <math>a_0</math> and q a factor of <math>a_n</math>.  It can be shown that if a rational root of P(x) exists, it must be of the form <math>\frac{p}{q}</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.
-====Descartes' Law of Signs====
+====Descarte's Law of Signs====
-By the Fundamental Theorem of Algebra, the maximum number of distinct factors (not all necessarily real) of a polynomial of degree n is n.  This tells us nothing about whether or not these roots are positive or negative.  Decartes' Rule of Signs says that for a polynomial P(x), 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 P(-x), or is less than that by a multiple of 2.
+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.  Descarte'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 Theorem===
-[[Binomial theorem]] can be very useful for factoring and expanding polynomials.
+The [[Binomial Theorem]] can be very useful for factoring and expanding polynomials.
-==Introductory Topics==
-===Multiplying and Dividing 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.
-===Synthetic Division===
+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.
-==Intermediate and Olympiad Topics==
+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.
-===Transforming Polynomials===
+==Intermediate Topics==
+*[[Complex numbers]]
+*[[Fundamental Theorem of Algebra]]
+*[[Roots of unity]]
-===Other Important Topics===
+==Olympiad Topics==
 * [[Vieta's formulas]]
 * [[Newton's identities]]
+* [[Newton sums]]
-==Other Resources==
-An extensive coverage of this topic is given in [http://www.artofproblemsolving.com/Resources/Papers/PolynomialsAK.pdf A Few Elementary Properties of Polynomials] by Adeel Khan.
 == See also ==
 * [[Algebra]]
+[[Category:Algebra]]
+[[Category:Polynomials]]
+[[Category:Definition]]

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