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From Wikipedia, the free encyclopedia

The graph of a polynomial function of degree 3.
The graph of a polynomial function of degree 3.

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2yz + 1.

Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

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Transcription

Hi, I’m Rob. Welcome to Math Antics. In this video, we’re going to learn about Polynomials. That’s a big math word for a really big concept in Algebra, so pay attention. Now before we can understand what polynomials are, we need to learn about what mathematicians call “terms”. In Algebra, terms are mathematical expressions that are made up of two different parts: a number part and a variable part. In a term, the number part and the variable part are multiplied together, but since multiplication is implied in Algebra, the two parts of a term are usually written right next to each other with no times symbol between them. The number part is pretty simple… it’s just a number, like 2 or 5 or 1.4 And the number part has an official name… it’s called the “coefficient”. Now there’s another cool math word that you can use to impress your friends at parties! [party music, crowd noise] …and then I said, “That’s not my wife… that’s my coefficient!” [silence / crickets chirping] The variable part of a term is a little more complicated. It can be made up of one or more variables that are raised to a power. Like… the variable part could be 'x squared'. That’s a variable raised to a power. Or, the variable part could be just ‘y’. If you remember what we learned in our last video, you’ll realize that that also qualifies as a variable raised to a power. ‘y’ is the same as ‘y’ to the 1st power. But since the exponent ‘1’ doesn’t change anything, we don’t need to actually show it. Or… the variable part of a term could be some tricky combination of variables that are raised to powers, like ‘x squared’ times ‘y squared’. …or ‘a’ times ‘b squared’ times ‘c cubed’. Terms can have any number of variables like that, but the good news is that most of the time, you’ll only need to deal with terms that have one variable. …or maybe two in complicated problems. Oh, and there’s one thing I should point out before we move on… if you have a term like 6y, even though it would be fine to do the multiplication the other way around and write y6, it’s conventional to always write the number part of the term first and the variable part of the term second. Okay, so that’s the basic idea of a term. But there’s a little more to terms that we’ll learn in a minute. First, let’s see how this basic idea of a term helps us understand the basic idea of a polynomial. A polynomial is a combination of many terms. It’s kind of like a chain of terms that are all linked together using addition or subtraction. The terms themselves contain multiplication, but each term in a polynomial must be joined by either addition or subtraction. And polynomials can be made from any number of terms joined together, but there are a few specific names that are used to describe polynomials with a certain number of terms. If there’s only one term (which isn’t really a chain) then we call it a “monomial” because the prefix “mono” means “one”. If there are just two terms, then we call it a “binomial” because the prefix “bi” means “two”, and if there are three terms, then we call it a “trinomial” since the prefix “tri” means “three”. Beyond three terms, we usually just say “polynomial” since “poly” means “many”, and in fact, it’s common to simply use the term “polynomial” even when there are just 2 or 3 terms. Okay, so that’s the basic idea of a polynomial. It’s a series of terms that are joined together by addition or subtraction. Now, let’s see a typical example of a polynomial that will help us learn a little more about terms: 3 ‘x squared’ plus ‘x’ minus 5 How many terms does this polynomial have? Well, based on what we’ve learned so far, you’re probably not quit sure. If the terms are the parts that are joined together by addition or subtraction, then this should have three terms, but it looks like there’s something missing with the last two terms. This middle term is missing its number part, and this last term is missing its variable part. That doesn’t seem to fit with our original definition of a term. What’s up with that? Well, the middle term is easy to explain. There really is a number part there, but it’s just ‘1’. Do you remember how ‘1’ is always a factor of any number? But, since multiplying by ‘1’ has no effect on a number or variable, we don’t need to show it. So, if you see a term in a polynomial that has only a variable part, you know that the number part (or coefficient) of that term is just ‘1’. Okay, but what about this last term that’s missing its variable part? Well, that’s a little trickier. Do you remember in our last video about exponents in Algebra, we learned that any number or variable that’s raised to the 0th power just equals ‘1’? That means we can think of this last term as having a variable ‘x’ that’s being raised to the 0th power. Since that would always just equal ‘1’, it’s not really a variable in the true sense of the word, and it has no effect on the value of the term. But it makes sense, especially if you remember the other rule from the last video. That rule says that any number raised to the 1st power is just itself, which helps us see that this middle term is basically the same as ‘1x’ raised to the 1st power. Now do you see the pattern? Each term has a number part and each term has a variable part that is raised to a power: 0, 1 and 2. But since ‘x’ to the ‘0’ is just ‘1’, and ‘x’ to the ‘1’ is just ‘x’, and anything multiplied by ‘1’ is just itself, the polynomial gets simplified so that it no longer looks exactly like the pattern it comes from. Oh, and this last term… the one that doesn’t have a truly variable part… it’s called a CONSTANT term because its value always stays the same. Alright… Now that you know what a Polynomial is, let’s talk about an important property of terms and polynomials called their “degree”. Now that might sound like the units we use to measure temperature or angles, but the degree we’re talking about here is different. The degree of a term is determined by the power of the variable part. For example, in this term, since the power of the variable is 4, we say that the degree of the term is 4, or that it’s a 4th degree term. And in this term, the power of the variable is 3, so it’s a 3rd degree term. Likewise, this would be a 2nd degree term and this would be a 1st degree term. Oh, and I suppose you could call a term with no variable part a “zero degree” term, but it’s usually just referred to as a “constant term”. Things are a little more complicated when you have terms with more that one variable. In that case, you add up the powers of each variable to get the degree of the term. Since the powers in this term are 3 and 2, it’s a 5th degree term because 3 + 2 = 5. Okay, but why do we care about the degree of terms? Well, it’s because polynomials are often referred to by the degree of their highest term. If a polynomial contains a 4th degree term (but no higher terms), then it’s called a “4th degree” polynomial. But if its highest term is only a 2nd degree term, then it’s called a “2nd degree” polynomial. Another reason that we care about the degree of the terms is that it helps us decide the arrangement of a polynomial. We arrange the terms in a polynomial in order from the highest degree to the lowest. …ya know, cuz, mathematicians like to keep things organized… [mumbeling] …nice… let’s see…double check… Perfect! For example, this polynomial (which has 5 terms) should be rearranged so that the highest degree term is on the left, and the lowest degree term is on the right. But of course, not every polynomial has a term of every degree. This is a 5th degree polynomial, but it only has 3 terms. We should still put them in order from highest to lowest, even though it has terms that are missing. So, the “4x to the fifth” should come first. And then the “minus 10x”. And finally, the “plus 8”. By the way, it’s totally fine for a polynomial to have “missing” terms like that. And it’s sometimes helpful to think of those missing terms as just having coefficients that are all zeros. If the coefficient of a term is zero, then the whole term has a value of zero so it wouldn’t effect the polynomial at all. And speaking of coefficients… What if we need to re-arrange this polynomial so that its terms are in order from highest degree to lowest degree? The highest degree term is ‘5x squared’ but before we just move it to the front of the polynomial, it’s important to notice that it’s got a minus sign in front of it. Normally when we see a minus sign, we think of subtraction, but when it comes to polynomials, it’s best to think of a minus sign as a NEGATIVE SIGN that means the term right after it has a negative value (or a negative coefficient). In fact, instead of thinking of a polynomial as having terms that are added OR subtracted, it’s best to think of ALL of the terms as being ADDED, but that each term has either a POSITIVE or a NEGATIVE coefficient which is determined by the operator right in front of that term. For example, if you have this Polynomial, you should treat it as if all of the terms are being added together, and use the sign that’s directly in front of each term to tell you if it’s a positive or a negative term. This first term has a coefficient of ‘negative 4’, so it’s a negative term. The next term has a coefficient of ‘positive 6’, so it’s positive. The next term has a coefficient of ‘negative 8’, so it’s negative. And the constant term is just ‘positive 2’. And recognizing positive and negative coefficients helps us a lot when rearranging polynomials that have a mixture of positive and negative terms like our example here. If you think of the negative sign in front of the ‘5x squared’ term as part of its coefficient, then you’ll realize that when we move it to the front of the polynomial, the negative sign has to come with it. It has to come with it because it’s really a NEGATIVE term. If we don’t bring the negative sign along with it, we’ll be changing it into a positive term which would actually change the value of the polynomial. And in addition to helping us re-arrange them, treating a polynomial as a combination of positive and negative terms will be very helpful when we need to simplify them, which just so happens to be the subject of our next basic Algebra video. Alright, we’ve learned a LOT about polynomials in this video, and if you’re a little overwhelmed, don’t worry… it might just take some time for it all to make sense. Remember, you can always re-watch this video a few times, and doing some of the practice problems will help it all sink in. As always, thanks for watching Math Antics, and I’ll see ya next time. Learn more at www.mathantics.com

Contents

Etymology

The word polynomial joins two diverse roots: the Greek poly, meaning "many," and the Latin nomen, or name. It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century.[1]

Notation and terminology

The x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value (its value is "indeterminate"). It is thus more correct to call it an "indeterminate".[citation needed] However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.

It is a common usage to use uppercase letters for the indeterminates and the corresponding lowercase letters for the variables (arguments) of the associated function.

A polynomial P in the indeterminate x is commonly denoted either as P or as P(x). Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) date from the time where the distinction between a polynomial and the associated function was unclear. Moreover the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let P(x) be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If a denotes a number, a variable, another polynomial, or, more generally any expression, then P(a) denotes, by convention, the result of substituting a for x in P. Thus, the polynomial P defines the function

which is the polynomial function associated to P. Frequently, when using this notation, one supposes that a is a number. However one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if a is a polynomial then P(a) is also a polynomial.

More specifically, when a is the indeterminate x, then the image of x by this function is the polynomial P itself (substituting x to x does not change anything). In other words,

which justifies formally the existence of two notations for the same polynomial.

Definition

A polynomial is an expression that can be built from constants and symbols called indeterminates or variables by means of addition, multiplication and exponentiation to a non-negative integer power. Two such expressions that may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication are considered as defining the same polynomial.

A polynomial in a single indeterminate x can always be written (or rewritten) in the form

where are constants and is the indeterminate. The word "indeterminate" means that represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function.

This can be expressed more concisely by using summation notation:

That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number—called the coefficient of the term[2]—and a finite number of indeterminates, raised to nonnegative integer powers. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any one term with nonzero coefficient. Because x = x1, the degree of an indeterminate without a written exponent is one.

A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial.[3] The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial, 0, (which has no terms at all) is generally treated as not defined (but see below).[4]

For example:

is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3.

Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.

Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. For higher degrees the specific names are not commonly used, although quartic polynomial (for degree four) and quintic polynomial (for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, in x2 + 2x + 1 the term 2x is a linear term in a quadratic polynomial.

The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined as negative (either −1 or −∞).[5] These conventions are useful when defining Euclidean division of polynomials. The zero polynomial is also unique in that it is the only polynomial having an infinite number of roots. The graph of the zero polynomial, f(x) = 0, is the X-axis.

In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all its non-zero terms have degree n. The zero polynomial is homogeneous, and, as homogeneous polynomial, its degree is undefined.[6] For example, x3y2 + 7x2y3 − 3x5 is homogeneous of degree 5. For more details, see homogeneous polynomial.

The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.[7]

Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0.[8] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial,[9] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. The term "quadrinomial" is occasionally used for a four-term polynomial.

A real polynomial is a polynomial with real coefficients. The argument of the polynomial is not necessarily so restricted, for instance the s-plane variable in Laplace transforms. A real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients.

A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. A polynomial with two indeterminates is called a bivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result, for instance, from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is common, also, to say simply "polynomials in x, y, and z", listing the indeterminates allowed.

The evaluation of a polynomial consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method:

Arithmetic

Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering, and combining of like terms.[8][10] For example, if

then

which can be simplified to

To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.[8] For example, if

then

which can be simplified to

Polynomial evaluation can be used to compute the remainder of polynomial division by a polynomial of degree one, because the remainder of the division of f(x) by (xa) is f(a); see the polynomial remainder theorem. This is more efficient than the usual algorithm of division when the quotient is not needed.

  • A sum of polynomials is a polynomial.[4]
  • A product of polynomials is a polynomial.[4]
  • A composition of two polynomials is a polynomial, which is obtained by substituting a variable of the first polynomial by the second polynomial.[4]
  • The derivative of the polynomial anxn + an−1xn−1 + ... + a2x2 + a1x + a0 is the polynomial nanxn−1 + (n − 1)an−1xn−2 + ... + 2a2x + a1. If the set of the coefficients does not contain the integers (for example if the coefficients are integers modulo some prime number p), then kak should be interpreted as the sum of ak with itself, k times. For example, over the integers modulo p, the derivative of the polynomial xp + 1 is the polynomial 0.[11]
  • A primitive integral or antiderivative of the polynomial anxn + an−1xn−1 + ⋅⋅⋅ + a2x2 + a1x + a0 is the polynomial anxn+1/(n + 1) + an−1xn/n + ⋅⋅⋅ + a2x3/3 + a1x2/2 + a0x + c, where c is an arbitrary constant. For instance, the antiderivatives of x2 + 1 have the form 1/3x3 + x + c.

As for the integers, two kinds of divisions are considered for the polynomials. The Euclidean division of polynomials that generalizes the Euclidean division of the integers. It results in two polynomials, a quotient and a remainder that are characterized by the following property of the polynomials: given two polynomials a and b such that b ≠ 0, there exists a unique pair of polynomials, q, the quotient, and r, the remainder, such that a = b q + r and degree(r) < degree(b) (here the polynomial zero is supposed to have a negative degree). By hand as well as with a computer, this division can be computed by the polynomial long division algorithm.[12]

All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree.[13] For example, the factored form of

is

over the integers and the reals and

over the complex numbers.

The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems.

A formal quotient of polynomials, that is, an algebraic fraction wherein the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, yields another polynomial. For example, x3/12 is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to (1/12)x3 and 1/12 is just a constant. When this expression is used as a term, its coefficient is therefore 1/12. For similar reasons, if complex coefficients are allowed, one may have a single term like (2 + 3i) x3; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term. The expression 1/(x2 + 1) is not a polynomial because it includes division by a non-constant polynomial. The expression (5 + y)x is not a polynomial, because it contains an indeterminate used as exponent.

Because subtraction can be replaced by addition of the opposite quantity, and because positive integer exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and indeterminates using only addition and multiplication.

Polynomial functions

A polynomial function is a function that can be defined by evaluating a polynomial. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial

that evaluates to for all x in the domain of f (here, n is a non-negative integer and a0, a1, a2, ..., an are constant coefficients).

Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function maps reals to reals.

For example, the function f, defined by

is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in

According to the definition of polynomial functions, there may be expressions that obviously are not polynomials but nevertheless define polynomial functions. An example is the expression which takes the same values as the polynomial on the interval , and thus both expressions define the same polynomial function on this interval.

Every polynomial function is continuous, smooth, and entire.

Graphs

A polynomial function in one real variable can be represented by a graph.

  • The graph of the zero polynomial
f(x) = 0
is the x-axis.
  • The graph of a degree 0 polynomial
f(x) = a0, where a0 ≠ 0,
is a horizontal line with y-intercept a0
  • The graph of a degree 1 polynomial (or linear function)
f(x) = a0 + a1x , where a1 ≠ 0,
is an oblique line with y-intercept a0 and slope a1.
  • The graph of a degree 2 polynomial
f(x) = a0 + a1x + a2x2, where a2 ≠ 0
is a parabola.
  • The graph of a degree 3 polynomial
f(x) = a0 + a1x + a2x2 + a3x3, where a3 ≠ 0
is a cubic curve.
  • The graph of any polynomial with degree 2 or greater
f(x) = a0 + a1x + a2x2 + ... + anxn , where an ≠ 0 and n ≥ 2
is a continuous non-linear curve.

A non-constant polynomial function tends to infinity when the variable increases indefinitely (in absolute value). If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction (one branch for positive x and one for negative x).

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

Equations

A polynomial equation, also called algebraic equation, is an equation of the form[14]

For example,

is a polynomial equation.

When considering equations, the indeterminates (variables) of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a polynomial identity like (x + y)(xy) = x2y2, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.

In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.

The number of real solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra.

Solving equations

Every polynomial P in x defines a function called the polynomial function associated to P; the equation P(x) = 0 is the polynomial equation associated to P. The solutions of this equation are called the roots of the polynomial, or the zeros of the associated function (they correspond to the points where the graph of the function meets the x-axis).

A number a is a root of a polynomial P if and only if the linear polynomial xa divides P, that is if there is another polynomial Q such that P = (xa) Q. It may happen that xa divides P more than once: if (xa)2 divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (xa)m divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots, as, with the above definitions, every number is a root of the zero polynomial, with an undefined multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.[15] The relation between the coefficients of a polynomial and its roots is described by Vieta's formulas.

Some polynomials, such as x2 + 1, do not have any roots among the real numbers. If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors xa, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree 1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of 2x – 1 = 0 is 1/2. Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expression; for example the golden ratio is the unique positive solution of In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving is to compute numerical approximations of the solutions.[16] There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).

For polynomials in more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations in several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations.

The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination.

A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm for solving them, and even for deciding whether the set of solutions is empty (see Hilbert's tenth problem). Some of the most famous problems that have been solved during the fifty last years are related to Diophantine equations, such as Fermat's Last Theorem.

Generalizations

There are several generalizations of the concept of polynomials.

Trigonometric polynomials

A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers.[17] The coefficients may be taken as real numbers, for real-valued functions.

If sin(nx) and cos(nx) are expanded in terms of sin(x) and cos(x), a trigonometric polynomial becomes a polynomial in the two variables sin(x) and cos(x) (using List of trigonometric identities#Multiple-angle formulae). Conversely, every polynomial in sin(x) and cos(x) may be converted, with Product-to-sum identities, into a linear combination of functions sin(nx) and cos(nx). This equivalence explains why linear combinations are called polynomials.

For complex coefficients, there is no difference between such a function and a finite Fourier series.

Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform.

Matrix polynomials

A matrix polynomial is a polynomial with square matrices as variables.[18] Given an ordinary, scalar-valued polynomial

this polynomial evaluated at a matrix A is

where I is the identity matrix.[19]

A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mn(R).

Laurent polynomials

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.

Rational functions

A rational fraction is the quotient (algebraic fraction) of two polynomials. Any algebraic expression that can be rewritten as a rational fraction is a rational function.

While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero.

The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate.

Power series

Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like irrational numbers cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge.

Other examples

  • A bivariate polynomial where the second variable is substituted by an exponential function applied to the first variable, for example P(x, ex), may be called an exponential polynomial.

Applications

Calculus

The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function.

Calculating derivatives and integrals of polynomial functions is particularly simple. For the polynomial function

the derivative with respect to x is

and the indefinite integral is

Abstract algebra

In abstract algebra, one distinguishes between polynomials and polynomial functions. A polynomial f in one indeterminate x over a ring R is defined as a formal expression of the form

where n is a natural number, the coefficients a0, . . ., an are elements of R, and x is a formal symbol, whose powers xi are just placeholders for the corresponding coefficients ai, so that the given formal expression is just a way to encode the sequence (a0, a1, . . .), where there is an n such that ai = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term aixi is interpreted as a polynomial that has zero coefficients at all powers of x other than xi. Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

   for all elements a, b of the ring R and all natural numbers k and l.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[x]. The map from R to R[x] sending r to rx0 is an injective homomorphism of rings, by which R is viewed as a subring of R[x]. If R is commutative, then R[x] is an algebra over R.

One can think of the ring R[x] as arising from R by adding one new element x to R, and extending in a minimal way to a ring in which x satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is xr = rx). To do this, one must add all powers of x and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring R[x] over the real numbers by factoring out the ideal of multiples of the polynomial x2 + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

If R is commutative, then one can associate to every polynomial P in R[x], a polynomial function f with domain and range equal to R (more generally one can take domain and range to be the same unital associative algebra over R). One obtains the value f(r) by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for x.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[x], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[x] such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R[x] and r is an element of R such that f(r) = 0, then the polynomial (xr) divides f. The converse is also true. The quotient can be computed using the polynomial long division.[20][21]

If F is a field and f and g are polynomials in F[x] with g ≠ 0, then there exist unique polynomials q and r in F[x] with

and such that the degree of r is smaller than the degree of g (using the convention that the polynomial 0 has a negative degree). The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain.

Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials. In the case of coefficients in a ring, "non-constant" must be replaced by "non-constant or non-unit" (both definitions agree in the case of coefficients in a field). Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit (and division of the unit factor by the same unit). When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials). These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein's criterion can also be used in some cases to determine irreducibility.

Other applications

Polynomials serve to approximate other functions,[22] such as the use of splines.

Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.

The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input.

History

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.

History of the notation

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.[23]

See also

Notes

  1. ^ See "polynomial" and "binomial", Compact Oxford English Dictionary
  2. ^ The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p.
  3. ^ This terminology dates from the time when the distinction was not clear between a polynomial and the function that it defines: a constant term and a constant polynomial define constant functions.
  4. ^ a b c d Barbeau, E.J. (2003). Polynomials. Springer. pp. 1–2. ISBN 978-0-387-40627-5.
  5. ^ Weisstein, Eric W. "Zero Polynomial". MathWorld.
  6. ^ In fact, as homogeneous function, it is homogeneous of every degree
  7. ^ Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 978-0-8176-3731-6.
  8. ^ a b c Edwards, Harold M. (1995). Linear Algebra. Springer. p. 47. ISBN 978-0-8176-3731-6.
  9. ^ Some authors use "monomial" to mean "monic monomial". See Knapp, Anthony W. (2007). Advanced Algebra: Along with a Companion Volume Basic Algebra. Springer. p. 457. ISBN 0-8176-4522-5.
  10. ^ Salomon, David (2006). Coding for Data and Computer Communications. Springer. p. 459. ISBN 978-0-387-23804-3.
  11. ^ Barbeau, E.J. (2003). Polynomials. Springer. pp. 64–5. ISBN 978-0-387-40627-5.
  12. ^ Peter H. Selby, Steve Slavin, Practical Algebra: A Self-Teaching Guide, 2nd Edition, Wiley, ISBN 0-471-53012-3 ISBN 978-0-471-53012-1
  13. ^ Barbeau, E.J. (2003). Polynomials. Springer. pp. 80–2. ISBN 978-0-387-40627-5.
  14. ^ Proskuryakov, I.V. (1994). "Algebraic equation". In Hazewinkel, Michiel. Encyclopaedia of Mathematics. vol. 1. Springer. ISBN 978-1-55608-010-4.
  15. ^ Leung,, Kam-tim; et al. (1992). Polynomials and Equations. Hong Kong University Press. p. 134. ISBN 9789622092716.
  16. ^ McNamee, J.M. (2007). Numerical Methods for Roots of Polynomials, Part 1. Elsevier. ISBN 978-0-08-048947-6.
  17. ^ Powell, Michael J. D. (1981). Approximation Theory and Methods. Cambridge University Press. ISBN 978-0-521-29514-7.
  18. ^ Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) [1982]. Matrix Polynomials. Classics in Applied Mathematics. 58. Lancaster, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-681-0. Zbl 1170.15300.
  19. ^ Horn & Johnson 1990, p. 36.
  20. ^ Irving, Ronald S. (2004). Integers, Polynomials, and Rings: A Course in Algebra. Springer. p. 129. ISBN 978-0-387-20172-6.
  21. ^ Jackson, Terrence H. (1995). From Polynomials to Sums of Squares. CRC Press. p. 143. ISBN 978-0-7503-0329-3.
  22. ^ de Villiers, Johann (2012). Mathematics of Approximation. Springer. ISBN 9789491216503.
  23. ^ Howard Eves, An Introduction to the History of Mathematics, Sixth Edition, Saunders, ISBN 0-03-029558-0

References

External links

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