Polynomial - Knowledge

2741: 2233: 2736:{\displaystyle {\begin{array}{rccrcrcrcr}{\color {Red}{P}}{\color {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{3y}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{5}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{5}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{5}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{5}}\cdot {\color {Blue}{1}})\end{array}}} 5849: 5679: 5753: 5654: 5806: 5896: 5948: 5708: 1604:. 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 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 1338: 222: 2977: 1146: 3211: 2746: 4792: 9157:, 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 6971:, 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. 1397:. 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 −∞). The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of 9089:

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

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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

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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

1333:{\displaystyle \underbrace {_{\,}3x^{2}} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {1} \end{smallmatrix}}\underbrace {-_{\,}5x} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {2} \end{smallmatrix}}\underbrace {+_{\,}4} _{\begin{smallmatrix}\mathrm {term} \\\mathrm {3} \end{smallmatrix}}.} 1612:, 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 also common to say simply "polynomials in 1047:

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 term with nonzero coefficient. Because

2228: 1538:, 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. Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a 4534: 2972:{\displaystyle {\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\end{array}}} 4091: 4505: 4311: 303:) dates from a time when 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 7698: 7532: 319:". 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. 5640: 5460:

of a polynomial is the computation of the corresponding polynomial function; that is, the evaluation consists of substituting a numerical value to each indeterminate and carrying out the indicated multiplications and additions.

7115:

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

8724:(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 7094:

provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see

6537: 2125: 860: 5056: 3206:{\displaystyle {\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}} 5139:

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

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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

9447: 3616: 1896: 5311: 5366: 2130: 3979: 1069:. 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). 1476:

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

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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. For example, if

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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

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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.

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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 it is to compute

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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

4787:{\displaystyle {\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\dots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}} 2099: 906: 322:

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

8229:, 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 7128:. 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 9973:

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

8116: 7039: 1710: 386: 8342:. 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 6588: 5471: 5206: 3452: 10121: 7164:

For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called

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A composition may be expanded to a sum of terms using the rules for multiplication and division of polynomials. The composition of two polynomials is another polynomial.

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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

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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

6990:. 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 5208:

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

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The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in

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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

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of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include

8791: 6635:, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. 3216: 5947: 1901: 9647: 1534:

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

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Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name.

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The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in

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The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called

2987: 1785: 7300: 5211: 6762:. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered. 8368:. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials (like 8766:, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the 5464:

For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using

1372:(for degree five) are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term 7745:

A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example

9091: 2751: 2238: 2223:{\displaystyle {\begin{aligned}\color {Red}P&\color {Red}{=2x+3y+5}\\\color {Blue}Q&\color {Blue}{=2x+5y+xy+1}\end{aligned}}} 7218:). Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as 5707: 6669:

The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the

3912: 9101:, 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 257:

represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.

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are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the

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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

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Any two such polynomials can be added, subtracted, or multiplied. Furthermore, the result in each case is another polynomial

8372:) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for 4850:, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient 5678: 1715: 121: 10510: 5316: 3403:

is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if

2038: 17: 10719: 10436: 8387: 7990: 1652: 10250: 10172: 10129: 10086: 9930: 9905: 9704: 9035: 8676: 7809:

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

3666: 353: 9126: 7210:. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general 6542: 5160: 4837:

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers

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of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any

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is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any

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is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the

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of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the

1445:. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, 1075: 10499: 10213: 8996: 1340:

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

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of the term – and a finite number of indeterminates, raised to non-negative integer powers.

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Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen

10452:

Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen

10260:

Mayr, K. (1937). "Über die Auflösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen".

4086:{\displaystyle 5(x-1)\left(x+{\frac {1+i{\sqrt {3}}}{2}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)} 10734: 10494: 8725: 7338: 6696: 3850: 865: 10579: 10186: 9755: 8343: 7350: 7215: 7188: 7000: 6674: 9141:, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in 10532: 9520:

Understanding Mathematics for Young Children: A Guide for Foundation Stage and Lower Primary Teachers

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That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero

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Rings, Fields, and Vector Spaces: An Introduction to Abstract Algebra Via Geometric Constructibility

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is an equality between two matrix polynomials, which holds for the specific matrices in question. A

4500:{\displaystyle na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\dots +2a_{2}x+a_{1}=\sum _{i=1}^{n}ia_{i}x^{i-1}.} 4306:{\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}} 10724: 10078: 9011: 8966: 8543: 7239: 7219: 7108: 6655: 3750: 3406: 498: 50: 46: 7044: 6375:

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

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

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A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a

10663: 8992: 8736: 7140: 6978:. One may want to express the solutions as explicit numbers; for example, the unique solution of 6663: 4108: 537:

that can be added and multiplied. Two polynomial expressions are considered as defining the same

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non-zero polynomials which cannot be factorized into the product of two non-constant polynomials

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cannot), but the rules for manipulating their terms are the same as for polynomials. Non-formal

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of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.

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Polynomials appear in many areas of mathematics and science. For example, they are used to form

10693: 10688: 10683: 10561: 9137: 9007: 8970: 8735:). These algorithms are not practicable for hand-written computation, but are available in any 8704: 8365: 8321: 8118:

So, most of the theory of the multivariate case can be reduced to an iterated univariate case.

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and integrals of polynomials is particularly simple, compared to other kinds of functions. The

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powers, and has a finite number of terms. An example of a polynomial of a single indeterminate

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denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.

8774:. As another example, in radix 5, a string of digits such as 132 denotes the (decimal) number 3380: 1573:

is a function from the reals to the reals that is defined by a real polynomial. Similarly, an

10673: 10653: 10428: 9661: 9631: 9615: 9497: 9391: 9373: 8478: 8200:

satisfies no other relations than the obligatory ones, plus commutation with all elements of

8162: 7890:. It is straightforward to verify that the polynomials in a given set of indeterminates over 7693:{\displaystyle P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},} 5455: 5371: 5135:

are constant coefficients). Generally, unless otherwise specified, polynomial functions have

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Polynomials of small degree have been given specific names. A polynomial of degree zero is a

550: 193: 9332: 7527:{\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},} 964: 10770: 10739: 10658: 10525: 10231: 9987: 9747: 9080: 9043: 9015: 8978: 7796: 7207: 7196: 7103:). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, 6991: 5092: 3375: 2113: 1356: 546: 542: 526: 326:

denotes a number, a variable, another polynomial, or, more generally, any expression, then

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to approximate other functions. In advanced mathematics, polynomials are used to construct

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also generalize polynomials, but the multiplication of two power series may not converge.

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When considering equations, the indeterminates (variables) of polynomials are also called

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A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields

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is, in general, too difficult to be done by hand-written computation. However, efficient

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is the real or complex numbers, whence the two concepts are not always distinguished in

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is not a polynomial, and it cannot be written as a finite sum of powers of the variable

10729: 10596: 10591: 10517: 10348: 10277: 9952: 9665: 9076: 8719: 8369: 8257: 7932: 7903: 7823: 7818: 7792: 7249: 7169: 7087: 6753: 6690: 6395: 6384: 5465: 4838: 4514: 3754: 3670: 3353: 1398: 935: 915: 401: 165: 149: 10465:"Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen. II" 8225:

Formation of the polynomial ring, together with forming factor rings by factoring out

7112: 5635:{\displaystyle (((((a_{n}x+a_{n-1})x+a_{n-2})x+\dotsb +a_{3})x+a_{2})x+a_{1})x+a_{0}.} 5404: 3749:. The quotient and remainder may be computed by any of several algorithms, including 10574: 10432: 10409: 10386: 10355: 10328: 10298: 10281: 10246: 10217: 10190: 10168: 10145: 10125: 10102: 10082: 10059: 10014: 9974: 9926: 9901: 9862: 9835: 9788: 9759: 9727: 9700: 9673: 9567: 9524: 9473: 9424: 9397: 9346: 9067:

is bounded by a polynomial function of some variable, such as the size of the input.

9039: 8140: 7845: 7800: 7784: 7778: 7368: 7362: 7346: 7104: 7091: 6643: 3838: 3638: 3628: 1566: 157: 9349: 9150: 8650:(using the convention that the polynomial 0 has a negative degree). The polynomials 10775: 10627: 10620: 10615: 10378: 10269: 10004: 9798: 9026:

Polynomials are frequently used to encode information about some other object. The

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instead of "roots". The study of the sets of zeros of polynomials is the object of

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are two polynomial expressions that represent the same polynomial; so, one has the

10749: 10448:"Ueber die Auflösung der algebraischen Gleichungen durch transcendente Functionen" 9588: 7214:

for solving them, or even for deciding whether the set of solutions is empty (see

7202:

A polynomial equation for which one is interested only in the solutions which are

5436:, and thus both expressions define the same polynomial function on this interval. 1419:

In the case of polynomials in more than one indeterminate, a polynomial is called

541:

if they may be transformed, one to the other, by applying the usual properties of

533:, but may be any expression that do not involve the indeterminates, and represent 10637: 10632: 10584: 10569: 10322: 10294: 10240: 10227: 10139: 10096: 10053: 9895: 9721: 9059: 8403: 7861: 7705: 7133: 5448: 5444: 3870: 3647: 1646: 1642: 1473: 1393:

The polynomial 0, which may be considered to have no terms at all, is called the

485:

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

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We find that the set of integers is not closed under this operation of division.

9273: 8246:, which proceeds similarly, starting out with the field of integers modulo some 8235:

over the real numbers by factoring out the ideal of multiples of the polynomial

1295: 1240: 1182: 10608: 10603: 10314: 10164: 9142: 9132: 8297: 7331: 7327: 7269: 7177: 7096: 6954: 6928: 6670: 6647: 6353: 6271: 5136: 4508: 3873:

the irreducible factors may have any degree. For example, the factored form of

3862: 3857:) also have a factored form in which the polynomial is written as a product of 1586: 522: 518: 141: 74: 70: 10406:

Tata Lectures on Theta II: Jacobian theta functions and differential equations

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

7372: 7342: 7323:). This equivalence explains why linear combinations are called polynomials. 7117: 5145: 3642:, depending on context. This is analogous to the fact that the ratio of two 1062: 1029:. Each term consists of the product of a number – called the 7291:), a trigonometric polynomial becomes a polynomial in the two variables sin( 7272:. The coefficients may be taken as real numbers, for real-valued functions. 6921:, counted with their respective multiplicities, cannot exceed the degree of 6532:{\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}=0.} 6349: 10744: 9956: 8247: 8243: 7849: 7180:

solutions, and, if this number is finite, for computing the solutions. See

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from a given domain is a polynomial function if there exists a polynomial

855:{\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0},} 9030:

of a matrix or linear operator contains information about the operator's

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The special case where all the polynomials are of degree one is called a

6950: 6658:

asserts that there can not exist a general formula in radicals. However,

5051:{\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}} 3866: 1558: 1364:. For higher degrees, the specific names are not commonly used, although 1030: 180: 66: 58: 38: 10273: 10205: 10098:

Solving Polynomial Equations: Foundations, Algorithms, and Applications

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records the simplest algebraic relation satisfied by that element. The

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In the ancient times, they succeeded only for degrees one and two. For

4130: 4126: 2101:

When polynomials are added together, the result is another polynomial.

3342:

As in the example, the product of polynomials is always a polynomial.

10678: 10216:, vol. 211 (Revised third ed.), New York: Springer-Verlag, 9354: 9300: 9064: 7211: 7173: 7172:. For a set of polynomial equations with several unknowns, there are 7148: 6357: 4104: 1547: 1482:", with the term of largest degree first, or in "ascending powers of 137: 129: 3869:, they have the degree either one or two. Over the integers and the 1058:, the degree of an indeterminate without a written exponent is one. 10668: 10033:

This paragraph assumes that the polynomials have coefficients in a

9503: 8984: 8974: 8782:

be a positive integer greater than 1. Then every positive integer

7788: 7152: 6957:, every non-constant polynomial has at least one root; this is the 6400: 6205: 6004:

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

1539: 514: 206: 145: 62: 10235:. This classical book covers most of the content of this article. 8897:{\displaystyle a=r_{m}b^{m}+r_{m-1}b^{m-1}+\dotsb +r_{1}b+r_{0},} 8763: 8296:. (More generally, one can take domain and range to be any same 7203: 7125: 6935:). The coefficients of a polynomial and its roots are related by 6639: 6356:). If the degree is higher than one, the graph does not have any 3643: 3335:{\displaystyle PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5.} 1578: 1026: 221: 161: 133: 10302: 9783:. Classics in Applied Mathematics. Vol. 58. Lancaster, PA: 6953:. If, however, the set of accepted solutions is expanded to the 10464: 10447: 10423:

Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007).

530: 8139:

to itself considered as a constant polynomial is an injective

7330:, there is no difference between such a function and a finite 10011:

Advanced Algebra: Along with a Companion Volume Basic Algebra

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is commutative, then one can associate with every polynomial

7719:

is a matrix polynomial equation which holds for all matrices

6141: 2028:{\displaystyle P+Q=(3x^{2}-3x^{2})+(-2x+3x)+5xy+4y^{2}+(8-2)} 8743:

can also be used in some cases to determine irreducibility.

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compute the factorization into irreducible polynomials (see

124:

to complicated scientific problems; they are used to define

4798:

is an arbitrary constant. For example, antiderivatives of

1351:. Polynomials of degree one, two or three are respectively 9779:

Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2009) .

7375:

as variables. Given an ordinary, scalar-valued polynomial

7337:

Trigonometric polynomials are widely used, for example in

1596:, a polynomial in more than one indeterminate is called a 932:

is the indeterminate. The word "indeterminate" means that

396:. Frequently, when using this notation, one supposes that 9809: 3650:, not necessarily an integer. For example, the fraction 3611:{\displaystyle (f\circ g)(x)=f(g(x))=(3x+2)^{2}+2(3x+2).} 120:, which encode a wide range of problems, from elementary 10547: 10377:. Undergraduate Texts in Mathematics. pp. 263–318. 9484:

This class of endomorphisms is closed under composition,

9344: 7301:

List of trigonometric identities#Multiple-angle formulae

7155:) polynomial equations of degree higher than 1,000 (see 6083:

The graph of a degree 1 polynomial (or linear function)

8542:. In this case, the quotient can be computed using the 6666:

of the roots of a polynomial expression of any degree.

10402:"Resolution of algebraic equations by theta constants" 9778: 9496:

Marecek, Lynn; Mathis, Andrea Honeycutt (6 May 2020).

9050:

counts the number of proper colourings of that graph.

3665:

For polynomials in one variable, there is a notion of

2743:

Carrying out the multiplication in each term produces

1891:{\displaystyle P+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8} 245:. When the polynomial is considered as an expression, 9831:

Integers, Polynomials, and Rings: A Course in Algebra

9324: 9322: 9320: 8794: 8608: 8518: 8481: 7993: 7935: 7906: 7544: 7381: 7047: 7003: 6545: 6409: 6277:

The graph of any polynomial with degree 2 or greater

5474: 5407: 5374: 5319: 5306:{\displaystyle f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.} 5214: 5163: 5064: 4934: 4537: 4517: 4323: 4139: 3982: 3915: 3879: 3504: 3460: 3409: 3383: 3356: 3219: 2985: 2749: 2236: 2128: 2041: 1904: 1788: 1718: 1655: 1149: 1078: 973: 938: 918: 868: 735: 651: 606: 559: 453: 356: 334:) denotes, by convention, the result of substituting 10422: 9643: 9621: 9605: 8194:, and extending in a minimal way to a ring in which 7884:

is a polynomial all of whose coefficients belong to

9385: 9383: 9381: 6931:roots are considered (this is a consequence of the 5361:{\displaystyle \left({\sqrt {1-x^{2}}}\right)^{2},} 2118:Polynomials can also be multiplied. To expand the 1600:. A polynomial with two indeterminates is called a 10507:"Euler's Investigations on the Roots of Equations" 10347: 10073:Beauregard, Raymond A.; Fraleigh, John B. (1973), 10072: 9723:Numerical Methods for Roots of Polynomials, Part 1 9317: 9196: 8995:locally looks like a polynomial function, and the 8960: 8896: 8762:In modern positional numbers systems, such as the 8630: 8530: 8496: 8110: 7976: 7921: 7799:that can be rewritten as a rational fraction is a 7692: 7526: 7078: 7033: 6875:is a nonzero polynomial, there is a highest power 6582: 6531: 5634: 5428: 5393: 5360: 5305: 5200: 5079: 5050: 4786: 4523: 4499: 4305: 4085: 3968: 3901: 3610: 3490: 3446: 3395: 3362: 3334: 3205: 2971: 2735: 2222: 2093: 2027: 1890: 1774: 1704: 1465:is homogeneous of degree 5. For more details, see 1332: 1100: 1017: 944: 924: 900: 854: 712: 634: 592: 477: 380: 9301:"Polynomials | Brilliant Math & Science Wiki" 6364:with vertical direction (one branch for positive 3837:. In this case, the quotient may be computed by 729:can always be written (or rewritten) in the form 10762: 9517:Haylock, Derek; Cockburn, Anne D. (2008-10-14). 9378: 5468:, which consists of rewriting the polynomial as 30:For less elementary aspects of the subject, see 10291:Introduction To Modern Algebra, Revised Edition 10116:Burden, Richard L.; Faires, J. Douglas (1993), 9516: 9131:The earliest known use of the equal sign is in 7176:to decide whether they have a finite number of 3865:, the irreducible factors are linear. Over the 1649:) and combining of like terms. For example, if 237:occurring in a polynomial is commonly called a 10570:Zero polynomial (degree undefined or −1 or −∞) 9785:Society for Industrial and Applied Mathematics 6915:. The number of roots of a nonzero polynomial 5368:which takes the same values as the polynomial 3801:asserts that the remainder of the division of 1592:A polynomial in one indeterminate is called a 963:This can be expressed more concisely by using 205:. That is, it means a sum of many terms (many 128:, which appear in settings ranging from basic 10533: 10462: 10445: 10095:Bronstein, Manuel; et al., eds. (2006). 9923:An Introduction to the History of Mathematics 9549: 9495: 7315:, into a linear combination of functions sin( 6352:when the variable increases indefinitely (in 4096:The computation of the factored form, called 10487: 10181:Horn, Roger A.; Johnson, Charles R. (1990). 10138:Cahen, Paul-Jean; Chabert, Jean-Luc (1997). 10137: 10115: 9659: 9452:. Yale University Press. 1965. p. 621. 9208: 9063:means that the time it takes to complete an 7233: 3969:{\displaystyle 5(x-1)\left(x^{2}+x+1\right)} 3370:of a single variable and another polynomial 1527:. The third term is a constant. Because the 1390:is a linear term in a quadratic polynomial. 553:of addition and multiplication. For example 10180: 9893: 9815: 9699:. Hong Kong University Press. p. 134. 9561: 9545: 9543: 9420:Coding for Data and Computer Communications 6817:. It may happen that a power (greater than 1636: 447:does not change anything). In other words, 311:) be a polynomial" is a shorthand for "let 280:). Formally, the name of the polynomial is 216: 10540: 10526: 10341: 9120: 8778:= 42. This representation is unique. Let 8216:). To do this, one must add all powers of 7740: 5144:to the reals, the resulting function is a 3673:of integers. This notion of the division 1561:coefficients. When it is used to define a 1546:, and a three-term polynomial is called a 191:, or "name". It was derived from the term 98:. An example with three indeterminates is 10094: 8712:. In the case of coefficients in a ring, 8618: 8242:. Another example is the construction of 7225: 6796:, that is if there is another polynomial 4918:a polynomial. More precisely, a function 1281: 1223: 1158: 1018:{\displaystyle \sum _{k=0}^{n}a_{k}x^{k}} 10320: 10158: 9564:Practical Algebra: A Self-Teaching Guide 9540: 9220: 4876:. For example, over the integers modulo 3841:, a special case of synthetic division. 220: 10399: 10051: 9854: 9719: 9627: 9611: 9562:Selby, Peter H.; Slavin, Steve (1991). 9416: 9389: 9369: 9328: 8222:and their linear combinations as well. 7303:). Conversely, every polynomial in sin( 7191:, for which another range of different 6961:. By successively dividing out factors 4899: 3849:All polynomials with coefficients in a 2104:Subtraction of polynomials is similar. 1628:", listing the indeterminates allowed. 723:A polynomial in a single indeterminate 61:, that involves only the operations of 14: 10763: 9827: 9746: 9465: 8786:can be expressed uniquely in the form 8751: 8584:, then there exist unique polynomials 7812: 7534:this polynomial evaluated at a matrix 6778: 6756:of the polynomial function defined by 1520:. In the second term, the coefficient 10521: 10308: 10288: 10262:Monatshefte für Mathematik und Physik 10238: 10008: 10003:Some authors use "monomial" to mean " 9881: 9692: 9586: 9442: 9440: 9345: 9271: 9244: 9232: 9021: 7772: 7356: 4914:is a function that can be defined by 3976:over the integers and the reals, and 2718: 2706: 2681: 2669: 2644: 2632: 2607: 2595: 2569: 2554: 2529: 2514: 2489: 2474: 2449: 2434: 2408: 2393: 2368: 2353: 2328: 2313: 2288: 2273: 2251: 2242: 2179: 2172: 2140: 2133: 1775:{\displaystyle Q=-3x^{2}+3x+4y^{2}+8} 713:{\displaystyle (x-1)(x-2)=x^{2}-3x+2} 315:be a polynomial in the indeterminate 10368: 10259: 10239:Leung, Kam-tim; et al. (1992). 10204: 9920: 9693:Leung, Kam-tim; et al. (1992). 9295: 9293: 9267: 9265: 7896:form a commutative ring, called the 7268:taking on the values of one or more 6684: 1542:, a two-term polynomial is called a 1401:. The graph of the zero polynomial, 229:of a polynomial function of degree 3 213:was first used in the 17th century. 9858:From Polynomials to Sums of Squares 9085:Abel–Ruffini theorem § History 7767: 7124:, two important branches of modern 7041:is the unique positive solution of 6709:of a nonzero univariate polynomial 6348:A non-constant polynomial function 6211:The graph of a degree 3 polynomial 6155:The graph of a degree 2 polynomial 6034:The graph of a degree 0 polynomial 4882:, the derivative of the polynomial 2094:{\displaystyle P+Q=x+5xy+4y^{2}+6.} 1641:Polynomials can be added using the 1500:is written in descending powers of 901:{\displaystyle a_{0},\ldots ,a_{n}} 529:power. The constants are generally 435:by this function is the polynomial 24: 10161:A First Course In Abstract Algebra 9644:Varberg, Purcell & Rigdon 2007 9437: 8991:, which roughly states that every 8388:Polynomial greatest common divisor 7855: 6949:, do not have any roots among the 6344:is a continuous non-linear curve. 1898:can be reordered and regrouped as 1308: 1305: 1302: 1299: 1253: 1250: 1247: 1244: 1195: 1192: 1189: 1186: 908:are constants that are called the 25: 10787: 10481: 10144:. American Mathematical Society. 9290: 9262: 9258:Compact Oxford English Dictionary 9256:See "polynomial" and "binomial", 8908:is a nonnegative integer and the 8111:{\displaystyle R=\left(R\right).} 7900:in these indeterminates, denoted 7034:{\displaystyle (1+{\sqrt {5}})/2} 6974:There may be several meanings of 6673:solutions are counted with their 6012:The graph of the zero polynomial 3667:Euclidean division of polynomials 2107: 1705:{\displaystyle P=3x^{2}-2x+5xy-2} 1569:is not so restricted. However, a 1506:. The first term has coefficient 1294: 1239: 1181: 1036: 9752:Approximation Theory and Methods 9469:Progress in Holomorphic Dynamics 9127:History of mathematical notation 6927:, and equals this degree if all 5946: 5894: 5847: 5804: 5751: 5706: 5677: 5652: 3853:(for example, the integers or a 3374:of any number of variables, the 253:defined by the polynomial, then 187:, meaning "many", and the Latin 160:, which are central concepts in 10027: 9997: 9980: 9967: 9914: 9887: 9875: 9848: 9821: 9772: 9740: 9713: 9686: 9653: 9580: 9555: 9510: 9489: 9459: 9410: 9363: 9338: 9197:Beauregard & Fraleigh (1973 9081:Quartic function § History 9055:computational complexity theory 8961:Interpolation and approximation 8746: 8381: 8290:with domain and range equal to 7829: 7283:) are expanded in terms of sin( 7195:exist, including the classical 4915: 3824: 2979:Combining similar terms yields 27:Type of mathematical expression 10245:. Hong Kong University Press. 9944: 9250: 9238: 9226: 9214: 9202: 9190: 8644:is smaller than the degree of 8102: 8089: 8081: 8043: 8029: 7997: 7971: 7939: 7916: 7910: 7554: 7548: 7391: 7385: 7182:System of polynomial equations 7020: 7004: 6959:fundamental theorem of algebra 6933:fundamental theorem of algebra 6701:Properties of polynomial roots 6679:fundamental theorem of algebra 5610: 5591: 5572: 5547: 5522: 5487: 5484: 5481: 5478: 5475: 5423: 5408: 5230: 5218: 5173: 5167: 5105:is a non-negative integer and 5074: 5068: 4861:understood to mean the sum of 4368: 4356: 3998: 3986: 3931: 3919: 3692:results in two polynomials, a 3602: 3587: 3572: 3556: 3550: 3547: 3541: 3535: 3526: 3520: 3517: 3505: 3470: 3464: 3419: 3413: 3345: 3186: 3168: 3111: 3093: 3060: 3024: 2726: 2702: 2692: 2665: 2655: 2628: 2618: 2591: 2577: 2550: 2540: 2510: 2500: 2470: 2460: 2430: 2416: 2389: 2379: 2349: 2339: 2309: 2299: 2269: 2022: 2010: 1976: 1955: 1949: 1917: 1108:is a term. The coefficient is 679: 667: 664: 652: 587: 575: 572: 560: 463: 457: 381:{\displaystyle a\mapsto P(a),} 372: 366: 360: 268:is commonly denoted either as 13: 1: 10730:Horner's method of evaluation 10214:Graduate Texts in Mathematics 10044: 9855:Jackson, Terrence H. (1995). 9466:Kriete, Hartje (1998-05-20). 9095: 9077:Cubic function § History 8450:if there exists a polynomial 8358:). This is not the case when 6583:{\displaystyle 3x^{2}+4x-5=0} 5439:Every polynomial function is 5201:{\displaystyle f(x)=x^{3}-x,} 3447:{\displaystyle f(x)=x^{2}+2x} 1631: 488: 10488:Markushevich, A.I. (2001) , 10321:Prasolov, Victor V. (2005). 9897:Mathematics of Approximation 9894:de Villiers, Johann (2012). 9670:Encyclopaedia of Mathematics 8732:Factorization of polynomials 8638:and such that the degree of 8392:Factorization of polynomials 7349:. They are also used in the 7079:{\displaystyle x^{2}-x-1=0.} 6734:. In other words, a root of 6378: 4906:Ring of polynomial functions 4511:(or indefinite integral) of 3844: 3799:polynomial remainder theorem 1130:is two, while the degree of 197:by replacing the Latin root 171: 7: 10735:Polynomial identity testing 10495:Encyclopedia of Mathematics 10408:. Springer. pp. 261–. 10404:. In Mumford, David (ed.). 10383:10.1007/978-3-030-75051-0_6 9660:Proskuryakov, I.V. (1994). 9390:Edwards, Harold M. (1995). 9172: 9092:Arithmetic in Nine Sections 8726:unique factorization domain 8662:are uniquely determined by 7929:in the univariate case and 7339:trigonometric interpolation 7151:allow solving easily (on a 6697:Root-finding of polynomials 4114: 3851:unique factorization domain 3621: 3213:which can be simplified to 1436:of its non-zero terms have 10: 10792: 10463:von Lindemann, F. (1892). 10446:von Lindemann, F. (1884). 10187:Cambridge University Press 10159:Fraleigh, John B. (1976), 10141:Integer-Valued Polynomials 10122:Prindle, Weber and Schmidt 10009:Knapp, Anthony W. (2007). 9925:(6th ed.). Saunders. 9861:. CRC Press. p. 143. 9828:Irving, Ronald S. (2004). 9756:Cambridge University Press 9472:. CRC Press. p. 159. 9124: 9074: 9070: 8999:, which states that every 8964: 8755: 8385: 8309:.) One obtains the value 8186:by adding one new element 8174:One can think of the ring 8149:is viewed as a subring of 7984:in the multivariate case. 7859: 7833: 7816: 7795:) of two polynomials. Any 7776: 7717:matrix polynomial identity 7713:matrix polynomial equation 7360: 7351:discrete Fourier transform 7237: 7189:system of linear equations 6942:Some polynomials, such as 6771:is a root of a polynomial 6694: 6688: 6677:. This fact is called the 6654:. For higher degrees, the 6590:is a polynomial equation. 6382: 6064:is a horizontal line with 5151:For example, the function 5148:that maps reals to reals. 4903: 4118: 4093:over the complex numbers. 2111: 1040: 635:{\displaystyle x^{2}-3x+2} 593:{\displaystyle (x-1)(x-2)} 29: 10707: 10646: 10559: 10342:Sethuraman, B.A. (1997). 10242:Polynomials and Equations 10163:(2nd ed.), Reading: 10013:. Springer. p. 457. 9834:. Springer. p. 129. 9696:Polynomials and Equations 9672:. Vol. 1. Springer. 9550:Marecek & Mathis 2020 9423:. Springer. p. 459. 9209:Burden & Faires (1993 9179:List of polynomial topics 9028:characteristic polynomial 8997:Stone–Weierstrass theorem 8912:s are integers such that 8679:, division with remainder 7313:Product-to-sum identities 7311:) may be converted, with 7234:Trigonometric polynomials 5644: 4121:Calculus with polynomials 3491:{\displaystyle g(x)=3x+2} 1112:, the indeterminates are 10371:"Polynomial Expressions" 10309:Moise, Edwin E. (1967), 10120:(5th ed.), Boston: 10079:Houghton Mifflin Company 9396:. Springer. p. 47. 9184: 9012:polynomial interpolation 8967:Polynomial interpolation 8685:and shows that the ring 8683:polynomial long division 8631:{\displaystyle f=q\,g+r} 8544:polynomial long division 8250:as the coefficient ring 7246:trigonometric polynomial 7240:Trigonometric polynomial 7141:numerical approximations 6664:numerical approximations 6123:is an oblique line with 4109:computer algebra systems 4102:polynomial factorization 3902:{\displaystyle 5x^{3}-5} 3751:polynomial long division 3396:{\displaystyle f\circ g} 1637:Addition and subtraction 1571:real polynomial function 1101:{\displaystyle -5x^{2}y} 419:More specifically, when 416:) is also a polynomial. 217:Notation and terminology 10720:Greatest common divisor 10375:Elements of Mathematics 10289:McCoy, Neal H. (1968), 9990:, it is homogeneous of 9816:Horn & Johnson 1990 9566:(2nd ed.). Wiley. 9499:Intermediate Algebra 2e 9449:Introduction to Algebra 9417:Salomon, David (2006). 9163:s denote constants and 9121:History of the notation 8993:differentiable function 8770:or base, in this case, 8737:computer algebra system 8705:irreducible polynomials 8497:{\displaystyle a\in R,} 8352:is the integers modulo 8344:Fermat's little theorem 7741:Exponential polynomials 7216:Hilbert's tenth problem 6660:root-finding algorithms 5953:Polynomial of degree 7: 5901:Polynomial of degree 6: 5854:Polynomial of degree 5: 5811:Polynomial of degree 4: 5758:Polynomial of degree 3: 5713:Polynomial of degree 2: 5684:Polynomial of degree 1: 5659:Polynomial of degree 0: 5394:{\displaystyle 1-x^{2}} 4507:Similarly, the general 3859:irreducible polynomials 2035:and then simplified to 1598:multivariate polynomial 912:of the polynomial, and 501:that can be built from 478:{\displaystyle P(x)=P,} 346:. Thus, the polynomial 181:joins two diverse roots 144:; and they are used in 47:mathematical expression 10592:Quadratic function (2) 10513:on September 24, 2012. 10052:Barbeau, E.J. (2003). 9720:McNamee, J.M. (2007). 9138:The Whetstone of Witte 8971:Orthogonal polynomials 8898: 8741:Eisenstein's criterion 8632: 8532: 8498: 8112: 7978: 7923: 7762:exponential polynomial 7694: 7580: 7528: 7417: 7226:Polynomial expressions 7158:Root-finding algorithm 7080: 7035: 6899:, which is called the 6642:, methods such as the 6584: 6533: 5636: 5430: 5395: 5362: 5307: 5202: 5081: 5052: 4788: 4740: 4525: 4501: 4464: 4307: 4282: 4107:are available in most 4087: 3970: 3903: 3612: 3492: 3448: 3397: 3364: 3336: 3207: 2973: 2737: 2224: 2095: 2029: 1892: 1776: 1706: 1467:Homogeneous polynomial 1368:(for degree four) and 1334: 1102: 1043:Degree of a polynomial 1019: 994: 946: 926: 902: 856: 714: 636: 594: 479: 382: 292:), but the use of the 230: 10575:Constant function (0) 10429:Pearson Prentice Hall 10400:Umemura, H. (2012) . 9921:Eves, Howard (1990). 9748:Powell, Michael J. D. 9593:mathworld.wolfram.com 9278:mathworld.wolfram.com 8899: 8776:1 × 5 + 3 × 5 + 2 × 5 8718:"non-constant or non- 8633: 8533: 8499: 8346:for an example where 8113: 7979: 7924: 7695: 7560: 7529: 7397: 7371:is a polynomial with 7220:Fermat's Last Theorem 7147:. The most efficient 7081: 7036: 6992:algebraic expressions 6976:"solving an equation" 6738:is a solution of the 6585: 6534: 6368:and one for negative 5637: 5431: 5396: 5363: 5308: 5203: 5082: 5053: 4789: 4720: 4526: 4502: 4444: 4308: 4262: 4088: 3971: 3904: 3775:and linear, that is, 3760:When the denominator 3613: 3493: 3449: 3398: 3365: 3337: 3208: 2974: 2738: 2225: 2112:Further information: 2096: 2030: 1893: 1777: 1707: 1594:univariate polynomial 1585:is a polynomial with 1577:is a polynomial with 1557:is a polynomial with 1357:quadratic polynomials 1335: 1103: 1041:Further information: 1020: 974: 947: 927: 903: 857: 715: 637: 595: 495:polynomial expression 480: 439:itself (substituting 423:is the indeterminate 408:is a polynomial then 404:). In particular, if 383: 350:defines the function 264:in the indeterminate 224: 10708:Tools and algorithms 10628:Quintic function (5) 10616:Quartic function (4) 10553:polynomial functions 10369:Toth, Gabor (2021). 9988:homogeneous function 9662:"Algebraic equation" 9523:. SAGE. p. 49. 9147:Arithemetica integra 9044:chromatic polynomial 8979:spline interpolation 8792: 8716:must be replaced by 8708:) can be defined as 8606: 8516: 8479: 7991: 7933: 7904: 7797:algebraic expression 7542: 7379: 7328:complex coefficients 7208:Diophantine equation 7197:Gaussian elimination 7109:Abel–Ruffini theorem 7045: 7001: 6662:may be used to find 6656:Abel–Ruffini theorem 6543: 6407: 5472: 5405: 5372: 5317: 5212: 5161: 5080:{\displaystyle f(x)} 5062: 4932: 4900:Polynomial functions 4535: 4515: 4321: 4137: 3980: 3913: 3877: 3634:rational expressions 3502: 3458: 3407: 3381: 3354: 3217: 2983: 2747: 2234: 2126: 2114:Polynomial expansion 2039: 1902: 1786: 1716: 1653: 1602:bivariate polynomial 1581:coefficients, and a 1147: 1076: 971: 936: 916: 866: 733: 649: 604: 557: 535:mathematical objects 527:non-negative integer 451: 354: 126:polynomial functions 118:polynomial equations 10638:Septic equation (7) 10633:Sextic equation (6) 10580:Linear function (1) 10311:Calculus: Complete 9666:Hazewinkel, Michiel 9587:Weisstein, Eric W. 9272:Weisstein, Eric W. 9099: 200 BCE 9001:continuous function 8758:Positional notation 8752:Positional notation 8571:are polynomials in 8531:{\displaystyle x-a} 8418:are polynomials in 8301:associative algebra 8283:polynomial function 7841:Formal power series 7836:Formal power series 7824:Laurent polynomials 7813:Laurent polynomials 7760:, may be called an 7145:continuous function 7088:quadratic equations 6994:; for example, the 6777:if and only if the 6740:polynomial equation 6391:polynomial equation 4912:polynomial function 4317:is the polynomial 3669:, generalizing the 3350:Given a polynomial 1353:linear polynomials, 1345:constant polynomial 1067:constant polynomial 958:polynomial function 505:and symbols called 390:polynomial function 294:functional notation 158:algebraic varieties 79:nonnegative integer 18:Polynomial notation 10604:Cubic function (3) 10597:Quadratic equation 10274:10.1007/BF01707992 10118:Numerical Analysis 9975:constant functions 9951:that are integers 9781:Matrix Polynomials 9347:Weisstein, Eric W. 9036:minimal polynomial 9022:Other applications 8894: 8677:Euclidean division 8628: 8528: 8494: 8424:, it is said that 8370:Euclidean division 8258:modular arithmetic 8108: 7974: 7919: 7846:irrational numbers 7819:Laurent polynomial 7793:algebraic fraction 7773:Rational functions 7690: 7524: 7357:Matrix polynomials 7347:periodic functions 7250:linear combination 7170:algebraic geometry 7076: 7031: 6691:Algebraic equation 6685:Solving equations 6580: 6529: 6396:algebraic equation 6385:Algebraic equation 6362:parabolic branches 5632: 5426: 5391: 5358: 5303: 5198: 5077: 5058:that evaluates to 5048: 4892:is the polynomial 4784: 4521: 4497: 4303: 4133:of the polynomial 4083: 3966: 3899: 3793:for some constant 3755:synthetic division 3671:Euclidean division 3639:rational functions 3629:rational fractions 3608: 3488: 3444: 3393: 3360: 3332: 3203: 3201: 2969: 2967: 2733: 2731: 2724: 2712: 2690: 2675: 2653: 2638: 2616: 2601: 2575: 2563: 2538: 2523: 2498: 2483: 2458: 2443: 2414: 2402: 2377: 2362: 2337: 2322: 2297: 2282: 2257: 2248: 2220: 2218: 2215: 2176: 2167: 2137: 2091: 2025: 1888: 1772: 1702: 1583:complex polynomial 1575:integer polynomial 1488:". The polynomial 1370:quintic polynomial 1366:quartic polynomial 1330: 1326: 1324: 1323: 1291: 1271: 1269: 1268: 1236: 1213: 1211: 1210: 1178: 1098: 1015: 965:summation notation 942: 922: 898: 852: 710: 632: 590: 475: 378: 231: 166:algebraic geometry 150:numerical analysis 10758: 10757: 10699:Quasi-homogeneous 10415:978-0-8176-4578-6 10392:978-3-030-75050-3 10361:978-0-387-94848-5 10334:978-3-642-04012-2 10223:978-0-387-95385-4 10196:978-0-521-38632-6 10151:978-0-8218-0388-2 10108:978-3-540-27357-8 10065:978-0-387-40627-5 10020:978-0-8176-4522-9 9868:978-0-7503-0329-3 9841:978-0-387-20172-6 9794:978-0-89871-681-8 9765:978-0-521-29514-7 9733:978-0-08-048947-6 9679:978-1-55608-010-4 9573:978-0-471-53012-1 9530:978-1-4462-0497-9 9479:978-0-582-32388-9 9430:978-0-387-23804-3 9403:978-0-8176-3731-6 9350:"Zero Polynomial" 9090:from the Chinese 9040:algebraic element 8702:(more correctly, 8700:prime polynomials 8674:. This is called 8155:. In particular, 8141:ring homomorphism 7977:{\displaystyle R} 7922:{\displaystyle R} 7801:rational function 7785:rational fraction 7779:Rational function 7369:matrix polynomial 7363:Matrix polynomial 7105:Niels Henrik Abel 7092:quadratic formula 7018: 6779:linear polynomial 6652:quartic equations 6644:quadratic formula 6393:, also called an 6350:tends to infinity 5998: 5941: 5889: 5842: 5799: 5746: 5701: 5672: 5343: 4782: 4687: 4655: 4617: 4579: 4524:{\displaystyle P} 4076: 4070: 4035: 4029: 3363:{\displaystyle f} 1362:cubic polynomials 1274: 1272: 1216: 1214: 1152: 1150: 945:{\displaystyle x} 925:{\displaystyle x} 16:(Redirected from 10783: 10621:Quartic equation 10542: 10535: 10528: 10519: 10518: 10514: 10509:. Archived from 10502: 10476: 10459: 10442: 10427:(9th ed.). 10419: 10396: 10365: 10353: 10338: 10317: 10305: 10285: 10256: 10234: 10200: 10177: 10155: 10134: 10112: 10091: 10069: 10038: 10031: 10025: 10024: 10001: 9995: 9984: 9978: 9971: 9965: 9963: 9948: 9937: 9936: 9918: 9912: 9911: 9891: 9885: 9879: 9873: 9872: 9852: 9846: 9845: 9825: 9819: 9813: 9807: 9806: 9776: 9770: 9769: 9744: 9738: 9737: 9717: 9711: 9710: 9690: 9684: 9683: 9657: 9651: 9641: 9635: 9625: 9619: 9609: 9603: 9602: 9600: 9599: 9589:"Ruffini's Rule" 9584: 9578: 9577: 9559: 9553: 9547: 9538: 9537: 9514: 9508: 9507: 9493: 9487: 9486: 9463: 9457: 9456: 9444: 9435: 9434: 9414: 9408: 9407: 9387: 9376: 9367: 9361: 9360: 9359: 9342: 9336: 9326: 9315: 9314: 9312: 9311: 9297: 9288: 9287: 9285: 9284: 9269: 9260: 9254: 9248: 9242: 9236: 9230: 9224: 9218: 9212: 9206: 9200: 9194: 9168: 9162: 9116: 9100: 9097: 8989:Taylor's theorem 8956: 8951:= 0, 1, . . . , 8945: 8929: 8903: 8901: 8900: 8895: 8890: 8889: 8874: 8873: 8855: 8854: 8839: 8838: 8820: 8819: 8810: 8809: 8777: 8773: 8693:Euclidean domain 8690: 8673: 8667: 8661: 8655: 8649: 8643: 8637: 8635: 8634: 8629: 8601: 8595: 8589: 8583: 8576: 8570: 8564: 8554: 8541: 8537: 8535: 8534: 8529: 8511: 8507: 8503: 8501: 8500: 8495: 8474: 8461: 8455: 8449: 8444:is a divisor of 8443: 8437: 8429: 8423: 8417: 8411: 8401: 8377: 8363: 8357: 8351: 8341: 8335: 8329: 8319: 8308: 8295: 8289: 8280: 8274: 8268: 8255: 8241: 8234: 8221: 8215: 8205: 8199: 8185: 8180:as arising from 8179: 8170: 8160: 8154: 8148: 8138: 8132: 8126: 8117: 8115: 8114: 8109: 8101: 8100: 8088: 8084: 8080: 8079: 8055: 8054: 8028: 8027: 8009: 8008: 7983: 7981: 7980: 7975: 7970: 7969: 7951: 7950: 7928: 7926: 7925: 7920: 7895: 7889: 7883: 7877:commutative ring 7874: 7768:Related concepts 7759: 7699: 7697: 7696: 7691: 7686: 7685: 7676: 7675: 7657: 7656: 7647: 7646: 7631: 7630: 7615: 7614: 7602: 7601: 7600: 7591: 7590: 7579: 7574: 7533: 7531: 7530: 7525: 7520: 7519: 7510: 7509: 7491: 7490: 7481: 7480: 7465: 7464: 7452: 7451: 7439: 7438: 7437: 7428: 7427: 7416: 7411: 7193:solution methods 7130:quintic function 7101:quartic equation 7085: 7083: 7082: 7077: 7057: 7056: 7040: 7038: 7037: 7032: 7027: 7019: 7014: 6989: 6985: 6970: 6948: 6937:Vieta's formulas 6926: 6920: 6914: 6908: 6898: 6892: 6880: 6874: 6868: 6858: 6853:, and otherwise 6852: 6842: 6837:; in this case, 6836: 6830: 6820: 6816: 6801: 6795: 6789: 6776: 6770: 6761: 6751: 6737: 6733: 6722: 6718: 6714: 6634: 6589: 6587: 6586: 6581: 6558: 6557: 6538: 6536: 6535: 6530: 6522: 6521: 6506: 6505: 6493: 6492: 6483: 6482: 6464: 6463: 6448: 6447: 6429: 6428: 6419: 6418: 6342: 6326: 6268: 6258: 6202: 6192: 6151: 6139: 6138: 6129: 6120: 6110: 6080: 6079: 6070: 6061: 6051: 6030: 6023: 5997: 5989: 5955: 5950: 5940: 5928: 5903: 5898: 5888: 5856: 5851: 5841: 5813: 5808: 5798: 5782: 5760: 5755: 5745: 5733: 5715: 5710: 5700: 5686: 5681: 5671: 5661: 5656: 5641: 5639: 5638: 5633: 5628: 5627: 5609: 5608: 5590: 5589: 5571: 5570: 5546: 5545: 5521: 5520: 5499: 5498: 5435: 5433: 5432: 5429:{\displaystyle } 5427: 5401:on the interval 5400: 5398: 5397: 5392: 5390: 5389: 5367: 5365: 5364: 5359: 5354: 5353: 5348: 5344: 5342: 5341: 5326: 5312: 5310: 5309: 5304: 5296: 5295: 5283: 5282: 5264: 5263: 5248: 5247: 5207: 5205: 5204: 5199: 5188: 5187: 5156: 5134: 5104: 5098: 5090: 5086: 5084: 5083: 5078: 5057: 5055: 5054: 5049: 5047: 5046: 5031: 5030: 5018: 5017: 5008: 5007: 4989: 4988: 4973: 4972: 4954: 4953: 4944: 4943: 4923: 4895: 4891: 4881: 4875: 4864: 4860: 4849: 4833: 4821: 4819: 4818: 4815: 4812: 4804: 4797: 4793: 4791: 4790: 4785: 4783: 4781: 4770: 4769: 4768: 4753: 4752: 4742: 4739: 4734: 4701: 4700: 4688: 4683: 4682: 4681: 4672: 4671: 4661: 4656: 4651: 4650: 4649: 4640: 4639: 4629: 4618: 4613: 4612: 4611: 4602: 4601: 4585: 4580: 4578: 4567: 4566: 4565: 4550: 4549: 4539: 4530: 4528: 4527: 4522: 4506: 4504: 4503: 4498: 4493: 4492: 4477: 4476: 4463: 4458: 4440: 4439: 4424: 4423: 4402: 4401: 4386: 4385: 4352: 4351: 4336: 4335: 4316: 4313:with respect to 4312: 4310: 4309: 4304: 4302: 4301: 4292: 4291: 4281: 4276: 4258: 4257: 4242: 4241: 4229: 4228: 4219: 4218: 4200: 4199: 4184: 4183: 4165: 4164: 4155: 4154: 4092: 4090: 4089: 4084: 4082: 4078: 4077: 4072: 4071: 4066: 4054: 4041: 4037: 4036: 4031: 4030: 4025: 4013: 3975: 3973: 3972: 3967: 3965: 3961: 3948: 3947: 3908: 3906: 3905: 3900: 3892: 3891: 3871:rational numbers 3836: 3822: 3811: 3796: 3792: 3770: 3748: 3736: 3719: 3705: 3691: 3661: 3657: 3617: 3615: 3614: 3609: 3580: 3579: 3497: 3495: 3494: 3489: 3453: 3451: 3450: 3445: 3434: 3433: 3402: 3400: 3399: 3394: 3373: 3369: 3367: 3366: 3361: 3341: 3339: 3338: 3333: 3316: 3315: 3297: 3296: 3269: 3268: 3241: 3240: 3212: 3210: 3209: 3204: 3202: 3160: 3159: 3137: 3136: 3118: 3117: 3082: 3081: 3016: 3015: 3002: 2978: 2976: 2975: 2970: 2968: 2913: 2912: 2895: 2894: 2872: 2871: 2837: 2836: 2816: 2815: 2780: 2779: 2766: 2742: 2740: 2739: 2734: 2732: 2725: 2723: 2713: 2711: 2691: 2689: 2676: 2674: 2654: 2652: 2639: 2637: 2617: 2615: 2602: 2600: 2584: 2583: 2576: 2574: 2564: 2562: 2539: 2537: 2524: 2522: 2499: 2497: 2484: 2482: 2459: 2457: 2444: 2442: 2423: 2422: 2415: 2413: 2403: 2401: 2378: 2376: 2363: 2361: 2338: 2336: 2323: 2321: 2298: 2296: 2283: 2281: 2267: 2265: 2258: 2256: 2249: 2247: 2229: 2227: 2226: 2221: 2219: 2214: 2166: 2100: 2098: 2097: 2092: 2084: 2083: 2034: 2032: 2031: 2026: 2006: 2005: 1948: 1947: 1932: 1931: 1897: 1895: 1894: 1889: 1881: 1880: 1856: 1855: 1813: 1812: 1781: 1779: 1778: 1773: 1765: 1764: 1740: 1739: 1711: 1709: 1708: 1703: 1674: 1673: 1627: 1621: 1536:distributive law 1526: 1525: 1519: 1515: 1510:, indeterminate 1509: 1505: 1499: 1487: 1481: 1464: 1444: 1443: 1431: 1430: 1411: 1389: 1378: 1339: 1337: 1336: 1331: 1325: 1320: 1311: 1292: 1287: 1283: 1282: 1270: 1265: 1256: 1237: 1232: 1225: 1224: 1212: 1207: 1198: 1179: 1174: 1173: 1172: 1160: 1159: 1139: 1135: 1129: 1124:, the degree of 1123: 1117: 1111: 1107: 1105: 1104: 1099: 1094: 1093: 1057: 1024: 1022: 1021: 1016: 1014: 1013: 1004: 1003: 993: 988: 951: 949: 948: 943: 931: 929: 928: 923: 907: 905: 904: 899: 897: 896: 878: 877: 861: 859: 858: 853: 848: 847: 832: 831: 819: 818: 809: 808: 790: 789: 774: 773: 755: 754: 745: 744: 728: 719: 717: 716: 711: 694: 693: 641: 639: 638: 633: 616: 615: 599: 597: 596: 591: 484: 482: 481: 476: 387: 385: 384: 379: 154:polynomial rings 112: 97: 86: 21: 10791: 10790: 10786: 10785: 10784: 10782: 10781: 10780: 10761: 10760: 10759: 10754: 10703: 10642: 10585:Linear equation 10555: 10546: 10505: 10484: 10479: 10439: 10416: 10393: 10362: 10335: 10295:Allyn and Bacon 10253: 10224: 10197: 10183:Matrix Analysis 10175: 10152: 10132: 10109: 10089: 10066: 10047: 10042: 10041: 10032: 10028: 10021: 10007:monomial". See 10002: 9998: 9985: 9981: 9972: 9968: 9959: 9949: 9945: 9940: 9933: 9919: 9915: 9908: 9892: 9888: 9880: 9876: 9869: 9853: 9849: 9842: 9826: 9822: 9814: 9810: 9795: 9777: 9773: 9766: 9745: 9741: 9734: 9718: 9714: 9707: 9691: 9687: 9680: 9658: 9654: 9642: 9638: 9626: 9622: 9610: 9606: 9597: 9595: 9585: 9581: 9574: 9560: 9556: 9548: 9541: 9531: 9515: 9511: 9494: 9490: 9480: 9464: 9460: 9446: 9445: 9438: 9431: 9415: 9411: 9404: 9388: 9379: 9368: 9364: 9343: 9339: 9327: 9318: 9309: 9307: 9299: 9298: 9291: 9282: 9280: 9270: 9263: 9255: 9251: 9243: 9239: 9231: 9227: 9219: 9215: 9207: 9203: 9195: 9191: 9187: 9175: 9164: 9158: 9129: 9123: 9102: 9098: 9087: 9075:Main articles: 9073: 9060:polynomial time 9024: 9014:and the use of 8981: 8963: 8947: 8940: 8931: 8924: 8915: 8885: 8881: 8869: 8865: 8844: 8840: 8828: 8824: 8815: 8811: 8805: 8801: 8793: 8790: 8789: 8775: 8772:4 × 10 + 5 × 10 8771: 8760: 8754: 8749: 8686: 8669: 8663: 8657: 8651: 8645: 8639: 8607: 8604: 8603: 8597: 8591: 8585: 8578: 8572: 8566: 8560: 8550: 8539: 8517: 8514: 8513: 8509: 8505: 8480: 8477: 8476: 8463: 8457: 8451: 8445: 8439: 8433: 8425: 8419: 8413: 8407: 8404:integral domain 8397: 8394: 8386:Main articles: 8384: 8373: 8359: 8353: 8347: 8337: 8331: 8330:for the symbol 8325: 8310: 8304: 8291: 8285: 8276: 8270: 8264: 8251: 8236: 8230: 8217: 8207: 8201: 8195: 8181: 8175: 8166: 8156: 8150: 8144: 8134: 8128: 8122: 8096: 8092: 8069: 8065: 8050: 8046: 8039: 8035: 8023: 8019: 8004: 8000: 7992: 7989: 7988: 7965: 7961: 7946: 7942: 7934: 7931: 7930: 7905: 7902: 7901: 7898:polynomial ring 7891: 7885: 7879: 7870: 7864: 7862:Polynomial ring 7858: 7856:Polynomial ring 7838: 7832: 7821: 7815: 7781: 7775: 7770: 7746: 7743: 7731: 7723:in a specified 7706:identity matrix 7681: 7677: 7671: 7667: 7652: 7648: 7642: 7638: 7626: 7622: 7610: 7606: 7596: 7592: 7586: 7582: 7581: 7575: 7564: 7543: 7540: 7539: 7515: 7511: 7505: 7501: 7486: 7482: 7476: 7472: 7460: 7456: 7447: 7443: 7433: 7429: 7423: 7419: 7418: 7412: 7401: 7380: 7377: 7376: 7373:square matrices 7365: 7359: 7341:applied to the 7270:natural numbers 7242: 7236: 7228: 7134:sextic equation 7113:Évariste Galois 7052: 7048: 7046: 7043: 7042: 7023: 7013: 7002: 6999: 6998: 6987: 6979: 6962: 6955:complex numbers 6943: 6922: 6916: 6910: 6904: 6894: 6882: 6876: 6870: 6864: 6854: 6848: 6838: 6832: 6822: 6818: 6803: 6797: 6791: 6781: 6772: 6766: 6757: 6742: 6735: 6724: 6720: 6716: 6710: 6703: 6693: 6687: 6609: 6553: 6549: 6544: 6541: 6540: 6517: 6513: 6501: 6497: 6488: 6484: 6478: 6474: 6453: 6449: 6437: 6433: 6424: 6420: 6414: 6410: 6408: 6405: 6404: 6387: 6381: 6343: 6336: 6328: 6322: 6310: 6300: 6293: 6279: 6269: 6266: 6260: 6254: 6244: 6234: 6227: 6213: 6203: 6200: 6194: 6188: 6178: 6171: 6157: 6150: 6144: 6137: 6131: 6125: 6124: 6122: 6118: 6112: 6106: 6099: 6085: 6078: 6072: 6066: 6065: 6063: 6059: 6053: 6050: 6036: 6026: 6024: 6014: 6002: 5999: 5991: 5990: 5956: 5954: 5951: 5942: 5930: 5929: 5904: 5902: 5899: 5890: 5882: 5857: 5855: 5852: 5843: 5839: 5814: 5812: 5809: 5800: 5784: 5783: 5761: 5759: 5756: 5747: 5735: 5734: 5716: 5714: 5711: 5702: 5687: 5685: 5682: 5673: 5662: 5660: 5657: 5647: 5623: 5619: 5604: 5600: 5585: 5581: 5566: 5562: 5535: 5531: 5510: 5506: 5494: 5490: 5473: 5470: 5469: 5466:Horner's method 5406: 5403: 5402: 5385: 5381: 5373: 5370: 5369: 5349: 5337: 5333: 5325: 5321: 5320: 5318: 5315: 5314: 5291: 5287: 5278: 5274: 5259: 5255: 5243: 5239: 5213: 5210: 5209: 5183: 5179: 5162: 5159: 5158: 5152: 5132: 5126: 5119: 5112: 5106: 5100: 5096: 5088: 5063: 5060: 5059: 5042: 5038: 5026: 5022: 5013: 5009: 5003: 4999: 4978: 4974: 4962: 4958: 4949: 4945: 4939: 4935: 4933: 4930: 4929: 4919: 4908: 4902: 4893: 4883: 4877: 4874: 4866: 4862: 4859: 4851: 4845: 4816: 4813: 4810: 4809: 4807: 4806: 4799: 4795: 4771: 4758: 4754: 4748: 4744: 4743: 4741: 4735: 4724: 4696: 4692: 4677: 4673: 4667: 4663: 4662: 4660: 4645: 4641: 4635: 4631: 4630: 4628: 4607: 4603: 4591: 4587: 4586: 4584: 4568: 4555: 4551: 4545: 4541: 4540: 4538: 4536: 4533: 4532: 4516: 4513: 4512: 4482: 4478: 4472: 4468: 4459: 4448: 4435: 4431: 4419: 4415: 4391: 4387: 4375: 4371: 4341: 4337: 4331: 4327: 4322: 4319: 4318: 4314: 4297: 4293: 4287: 4283: 4277: 4266: 4253: 4249: 4237: 4233: 4224: 4220: 4214: 4210: 4189: 4185: 4173: 4169: 4160: 4156: 4150: 4146: 4138: 4135: 4134: 4123: 4117: 4065: 4055: 4053: 4046: 4042: 4024: 4014: 4012: 4005: 4001: 3981: 3978: 3977: 3943: 3939: 3938: 3934: 3914: 3911: 3910: 3887: 3883: 3878: 3875: 3874: 3863:complex numbers 3847: 3827: 3813: 3802: 3794: 3776: 3761: 3738: 3721: 3710: 3696: 3674: 3659: 3651: 3648:rational number 3624: 3575: 3571: 3503: 3500: 3499: 3459: 3456: 3455: 3429: 3425: 3408: 3405: 3404: 3382: 3379: 3378: 3371: 3355: 3352: 3351: 3348: 3311: 3307: 3292: 3288: 3264: 3260: 3236: 3232: 3218: 3215: 3214: 3200: 3199: 3194: 3189: 3166: 3161: 3155: 3151: 3143: 3138: 3132: 3128: 3123: 3115: 3114: 3091: 3086: 3077: 3073: 3068: 3063: 3022: 3017: 3011: 3007: 3001: 2996: 2986: 2984: 2981: 2980: 2966: 2965: 2960: 2955: 2944: 2939: 2931: 2926: 2918: 2910: 2909: 2901: 2896: 2890: 2886: 2878: 2873: 2867: 2863: 2858: 2853: 2842: 2834: 2833: 2825: 2820: 2811: 2807: 2802: 2797: 2786: 2781: 2775: 2771: 2765: 2760: 2750: 2748: 2745: 2744: 2730: 2729: 2719: 2717: 2707: 2705: 2700: 2695: 2682: 2680: 2670: 2668: 2663: 2658: 2645: 2643: 2633: 2631: 2626: 2621: 2608: 2606: 2596: 2594: 2589: 2581: 2580: 2570: 2568: 2555: 2553: 2548: 2543: 2530: 2528: 2515: 2513: 2508: 2503: 2490: 2488: 2475: 2473: 2468: 2463: 2450: 2448: 2435: 2433: 2428: 2420: 2419: 2409: 2407: 2394: 2392: 2387: 2382: 2369: 2367: 2354: 2352: 2347: 2342: 2329: 2327: 2314: 2312: 2307: 2302: 2289: 2287: 2274: 2272: 2266: 2261: 2259: 2252: 2250: 2243: 2241: 2237: 2235: 2232: 2231: 2217: 2216: 2180: 2177: 2169: 2168: 2141: 2138: 2129: 2127: 2124: 2123: 2116: 2110: 2079: 2075: 2040: 2037: 2036: 2001: 1997: 1943: 1939: 1927: 1923: 1903: 1900: 1899: 1876: 1872: 1851: 1847: 1808: 1804: 1787: 1784: 1783: 1760: 1756: 1735: 1731: 1717: 1714: 1713: 1669: 1665: 1654: 1651: 1650: 1647:commutative law 1643:associative law 1639: 1634: 1623: 1613: 1555:real polynomial 1523: 1521: 1517: 1516:, and exponent 1511: 1507: 1501: 1489: 1483: 1477: 1474:commutative law 1446: 1439: 1437: 1426: 1424: 1402: 1395:zero polynomial 1380: 1373: 1322: 1321: 1316: 1313: 1312: 1298: 1293: 1280: 1276: 1275: 1273: 1267: 1266: 1261: 1258: 1257: 1243: 1238: 1222: 1218: 1217: 1215: 1209: 1208: 1203: 1200: 1199: 1185: 1180: 1168: 1164: 1157: 1154: 1153: 1151: 1148: 1145: 1144: 1137: 1131: 1125: 1119: 1113: 1109: 1089: 1085: 1077: 1074: 1073: 1049: 1045: 1039: 1009: 1005: 999: 995: 989: 978: 972: 969: 968: 937: 934: 933: 917: 914: 913: 892: 888: 873: 869: 867: 864: 863: 843: 839: 827: 823: 814: 810: 804: 800: 779: 775: 763: 759: 750: 746: 740: 736: 734: 731: 730: 724: 689: 685: 650: 647: 646: 611: 607: 605: 602: 601: 558: 555: 554: 491: 452: 449: 448: 355: 352: 351: 219: 201:with the Greek 174: 99: 88: 82: 35: 32:Polynomial ring 28: 23: 22: 15: 12: 11: 5: 10789: 10779: 10778: 10773: 10756: 10755: 10753: 10752: 10747: 10742: 10737: 10732: 10727: 10722: 10717: 10711: 10709: 10705: 10704: 10702: 10701: 10696: 10691: 10686: 10681: 10676: 10671: 10666: 10661: 10656: 10650: 10648: 10644: 10643: 10641: 10640: 10635: 10630: 10625: 10624: 10623: 10613: 10612: 10611: 10609:Cubic equation 10601: 10600: 10599: 10589: 10588: 10587: 10577: 10572: 10566: 10564: 10557: 10556: 10545: 10544: 10537: 10530: 10522: 10516: 10515: 10503: 10483: 10482:External links 10480: 10478: 10477: 10460: 10443: 10438:978-0131469686 10437: 10420: 10414: 10397: 10391: 10366: 10360: 10339: 10333: 10318: 10315:Addison-Wesley 10306: 10286: 10257: 10251: 10236: 10222: 10202: 10195: 10178: 10173: 10165:Addison-Wesley 10156: 10150: 10135: 10130: 10113: 10107: 10092: 10087: 10070: 10064: 10048: 10046: 10043: 10040: 10039: 10026: 10019: 9996: 9986:In fact, as a 9979: 9966: 9942: 9941: 9939: 9938: 9931: 9913: 9906: 9886: 9874: 9867: 9847: 9840: 9820: 9808: 9793: 9771: 9764: 9739: 9732: 9712: 9705: 9685: 9678: 9652: 9636: 9620: 9604: 9579: 9572: 9554: 9539: 9529: 9509: 9488: 9478: 9458: 9436: 9429: 9409: 9402: 9393:Linear Algebra 9377: 9362: 9337: 9316: 9289: 9261: 9249: 9237: 9235:, p. 190) 9225: 9223:, p. 245) 9221:Fraleigh (1976 9213: 9201: 9199:, p. 153) 9188: 9186: 9183: 9182: 9181: 9174: 9171: 9151:René Descartes 9143:Michael Stifel 9133:Robert Recorde 9125:Main article: 9122: 9119: 9072: 9069: 9023: 9020: 8962: 8959: 8936: 8920: 8893: 8888: 8884: 8880: 8877: 8872: 8868: 8864: 8861: 8858: 8853: 8850: 8847: 8843: 8837: 8834: 8831: 8827: 8823: 8818: 8814: 8808: 8804: 8800: 8797: 8764:decimal system 8756:Main article: 8753: 8750: 8748: 8745: 8714:"non-constant" 8627: 8624: 8621: 8617: 8614: 8611: 8527: 8524: 8521: 8493: 8490: 8487: 8484: 8383: 8380: 8107: 8104: 8099: 8095: 8091: 8087: 8083: 8078: 8075: 8072: 8068: 8064: 8061: 8058: 8053: 8049: 8045: 8042: 8038: 8034: 8031: 8026: 8022: 8018: 8015: 8012: 8007: 8003: 7999: 7996: 7973: 7968: 7964: 7960: 7957: 7954: 7949: 7945: 7941: 7938: 7918: 7915: 7912: 7909: 7860:Main article: 7857: 7854: 7834:Main article: 7831: 7828: 7817:Main article: 7814: 7811: 7777:Main article: 7774: 7771: 7769: 7766: 7742: 7739: 7729: 7689: 7684: 7680: 7674: 7670: 7666: 7663: 7660: 7655: 7651: 7645: 7641: 7637: 7634: 7629: 7625: 7621: 7618: 7613: 7609: 7605: 7599: 7595: 7589: 7585: 7578: 7573: 7570: 7567: 7563: 7559: 7556: 7553: 7550: 7547: 7523: 7518: 7514: 7508: 7504: 7500: 7497: 7494: 7489: 7485: 7479: 7475: 7471: 7468: 7463: 7459: 7455: 7450: 7446: 7442: 7436: 7432: 7426: 7422: 7415: 7410: 7407: 7404: 7400: 7396: 7393: 7390: 7387: 7384: 7361:Main article: 7358: 7355: 7332:Fourier series 7238:Main article: 7235: 7232: 7227: 7224: 7097:cubic equation 7075: 7072: 7069: 7066: 7063: 7060: 7055: 7051: 7030: 7026: 7022: 7017: 7012: 7009: 7006: 6689:Main article: 6686: 6683: 6638:In elementary 6579: 6576: 6573: 6570: 6567: 6564: 6561: 6556: 6552: 6548: 6528: 6525: 6520: 6516: 6512: 6509: 6504: 6500: 6496: 6491: 6487: 6481: 6477: 6473: 6470: 6467: 6462: 6459: 6456: 6452: 6446: 6443: 6440: 6436: 6432: 6427: 6423: 6417: 6413: 6383:Main article: 6380: 6377: 6354:absolute value 6346: 6345: 6332: 6318: 6308: 6298: 6291: 6278: 6275: 6264: 6252: 6242: 6232: 6225: 6212: 6209: 6198: 6186: 6176: 6169: 6156: 6153: 6148: 6135: 6116: 6104: 6097: 6084: 6081: 6076: 6057: 6048: 6035: 6032: 6013: 6001: 6000: 5952: 5945: 5943: 5900: 5893: 5891: 5853: 5846: 5844: 5810: 5803: 5801: 5757: 5750: 5748: 5712: 5705: 5703: 5683: 5676: 5674: 5658: 5651: 5648: 5646: 5643: 5631: 5626: 5622: 5618: 5615: 5612: 5607: 5603: 5599: 5596: 5593: 5588: 5584: 5580: 5577: 5574: 5569: 5565: 5561: 5558: 5555: 5552: 5549: 5544: 5541: 5538: 5534: 5530: 5527: 5524: 5519: 5516: 5513: 5509: 5505: 5502: 5497: 5493: 5489: 5486: 5483: 5480: 5477: 5425: 5422: 5419: 5416: 5413: 5410: 5388: 5384: 5380: 5377: 5357: 5352: 5347: 5340: 5336: 5332: 5329: 5324: 5302: 5299: 5294: 5290: 5286: 5281: 5277: 5273: 5270: 5267: 5262: 5258: 5254: 5251: 5246: 5242: 5238: 5235: 5232: 5229: 5226: 5223: 5220: 5217: 5197: 5194: 5191: 5186: 5182: 5178: 5175: 5172: 5169: 5166: 5130: 5124: 5117: 5110: 5076: 5073: 5070: 5067: 5045: 5041: 5037: 5034: 5029: 5025: 5021: 5016: 5012: 5006: 5002: 4998: 4995: 4992: 4987: 4984: 4981: 4977: 4971: 4968: 4965: 4961: 4957: 4952: 4948: 4942: 4938: 4901: 4898: 4870: 4855: 4805:have the form 4780: 4777: 4774: 4767: 4764: 4761: 4757: 4751: 4747: 4738: 4733: 4730: 4727: 4723: 4719: 4716: 4713: 4710: 4707: 4704: 4699: 4695: 4691: 4686: 4680: 4676: 4670: 4666: 4659: 4654: 4648: 4644: 4638: 4634: 4627: 4624: 4621: 4616: 4610: 4606: 4600: 4597: 4594: 4590: 4583: 4577: 4574: 4571: 4564: 4561: 4558: 4554: 4548: 4544: 4520: 4509:antiderivative 4496: 4491: 4488: 4485: 4481: 4475: 4471: 4467: 4462: 4457: 4454: 4451: 4447: 4443: 4438: 4434: 4430: 4427: 4422: 4418: 4414: 4411: 4408: 4405: 4400: 4397: 4394: 4390: 4384: 4381: 4378: 4374: 4370: 4367: 4364: 4361: 4358: 4355: 4350: 4347: 4344: 4340: 4334: 4330: 4326: 4300: 4296: 4290: 4286: 4280: 4275: 4272: 4269: 4265: 4261: 4256: 4252: 4248: 4245: 4240: 4236: 4232: 4227: 4223: 4217: 4213: 4209: 4206: 4203: 4198: 4195: 4192: 4188: 4182: 4179: 4176: 4172: 4168: 4163: 4159: 4153: 4149: 4145: 4142: 4119:Main article: 4116: 4113: 4081: 4075: 4069: 4064: 4061: 4058: 4052: 4049: 4045: 4040: 4034: 4028: 4023: 4020: 4017: 4011: 4008: 4004: 4000: 3997: 3994: 3991: 3988: 3985: 3964: 3960: 3957: 3954: 3951: 3946: 3942: 3937: 3933: 3930: 3927: 3924: 3921: 3918: 3898: 3895: 3890: 3886: 3882: 3846: 3843: 3839:Ruffini's rule 3743:) < degree( 3623: 3620: 3607: 3604: 3601: 3598: 3595: 3592: 3589: 3586: 3583: 3578: 3574: 3570: 3567: 3564: 3561: 3558: 3555: 3552: 3549: 3546: 3543: 3540: 3537: 3534: 3531: 3528: 3525: 3522: 3519: 3516: 3513: 3510: 3507: 3487: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3463: 3443: 3440: 3437: 3432: 3428: 3424: 3421: 3418: 3415: 3412: 3392: 3389: 3386: 3359: 3347: 3344: 3331: 3328: 3325: 3322: 3319: 3314: 3310: 3306: 3303: 3300: 3295: 3291: 3287: 3284: 3281: 3278: 3275: 3272: 3267: 3263: 3259: 3256: 3253: 3250: 3247: 3244: 3239: 3235: 3231: 3228: 3225: 3222: 3198: 3195: 3193: 3190: 3188: 3185: 3182: 3179: 3176: 3173: 3170: 3167: 3165: 3162: 3158: 3154: 3150: 3147: 3144: 3142: 3139: 3135: 3131: 3127: 3124: 3122: 3119: 3116: 3113: 3110: 3107: 3104: 3101: 3098: 3095: 3092: 3090: 3087: 3085: 3080: 3076: 3072: 3069: 3067: 3064: 3062: 3059: 3056: 3053: 3050: 3047: 3044: 3041: 3038: 3035: 3032: 3029: 3026: 3023: 3021: 3018: 3014: 3010: 3006: 3003: 3000: 2997: 2995: 2992: 2989: 2988: 2964: 2961: 2959: 2956: 2954: 2951: 2948: 2945: 2943: 2940: 2938: 2935: 2932: 2930: 2927: 2925: 2922: 2919: 2917: 2914: 2911: 2908: 2905: 2902: 2900: 2897: 2893: 2889: 2885: 2882: 2879: 2877: 2874: 2870: 2866: 2862: 2859: 2857: 2854: 2852: 2849: 2846: 2843: 2841: 2838: 2835: 2832: 2829: 2826: 2824: 2821: 2819: 2814: 2810: 2806: 2803: 2801: 2798: 2796: 2793: 2790: 2787: 2785: 2782: 2778: 2774: 2770: 2767: 2764: 2761: 2759: 2756: 2753: 2752: 2728: 2722: 2716: 2710: 2704: 2701: 2699: 2696: 2694: 2688: 2685: 2679: 2673: 2667: 2664: 2662: 2659: 2657: 2651: 2648: 2642: 2636: 2630: 2627: 2625: 2622: 2620: 2614: 2611: 2605: 2599: 2593: 2590: 2588: 2585: 2582: 2579: 2573: 2567: 2561: 2558: 2552: 2549: 2547: 2544: 2542: 2536: 2533: 2527: 2521: 2518: 2512: 2509: 2507: 2504: 2502: 2496: 2493: 2487: 2481: 2478: 2472: 2469: 2467: 2464: 2462: 2456: 2453: 2447: 2441: 2438: 2432: 2429: 2427: 2424: 2421: 2418: 2412: 2406: 2400: 2397: 2391: 2388: 2386: 2383: 2381: 2375: 2372: 2366: 2360: 2357: 2351: 2348: 2346: 2343: 2341: 2335: 2332: 2326: 2320: 2317: 2311: 2308: 2306: 2303: 2301: 2295: 2292: 2286: 2280: 2277: 2271: 2268: 2264: 2260: 2255: 2246: 2240: 2239: 2213: 2210: 2207: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2183: 2178: 2175: 2171: 2170: 2165: 2162: 2159: 2156: 2153: 2150: 2147: 2144: 2139: 2136: 2132: 2131: 2109: 2108:Multiplication 2106: 2090: 2087: 2082: 2078: 2074: 2071: 2068: 2065: 2062: 2059: 2056: 2053: 2050: 2047: 2044: 2024: 2021: 2018: 2015: 2012: 2009: 2004: 2000: 1996: 1993: 1990: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1946: 1942: 1938: 1935: 1930: 1926: 1922: 1919: 1916: 1913: 1910: 1907: 1887: 1884: 1879: 1875: 1871: 1868: 1865: 1862: 1859: 1854: 1850: 1846: 1843: 1840: 1837: 1834: 1831: 1828: 1825: 1822: 1819: 1816: 1811: 1807: 1803: 1800: 1797: 1794: 1791: 1771: 1768: 1763: 1759: 1755: 1752: 1749: 1746: 1743: 1738: 1734: 1730: 1727: 1724: 1721: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1680: 1677: 1672: 1668: 1664: 1661: 1658: 1638: 1635: 1633: 1630: 1589:coefficients. 1347:, or simply a 1329: 1319: 1315: 1314: 1310: 1307: 1304: 1301: 1297: 1296: 1290: 1286: 1279: 1264: 1260: 1259: 1255: 1252: 1249: 1246: 1242: 1241: 1235: 1231: 1228: 1221: 1206: 1202: 1201: 1197: 1194: 1191: 1188: 1184: 1183: 1177: 1171: 1167: 1163: 1156: 1097: 1092: 1088: 1084: 1081: 1038: 1037:Classification 1035: 1012: 1008: 1002: 998: 992: 987: 984: 981: 977: 941: 921: 895: 891: 887: 884: 881: 876: 872: 851: 846: 842: 838: 835: 830: 826: 822: 817: 813: 807: 803: 799: 796: 793: 788: 785: 782: 778: 772: 769: 766: 762: 758: 753: 749: 743: 739: 709: 706: 703: 700: 697: 692: 688: 684: 681: 678: 675: 672: 669: 666: 663: 660: 657: 654: 631: 628: 625: 622: 619: 614: 610: 589: 586: 583: 580: 577: 574: 571: 568: 565: 562: 551:distributivity 523:exponentiation 519:multiplication 511:indeterminates 490: 487: 474: 471: 468: 465: 462: 459: 456: 392:associated to 377: 374: 371: 368: 365: 362: 359: 218: 215: 173: 170: 142:social science 75:exponentiation 71:multiplication 51:indeterminates 49:consisting of 26: 9: 6: 4: 3: 2: 10788: 10777: 10774: 10772: 10769: 10768: 10766: 10751: 10750:Gröbner basis 10748: 10746: 10743: 10741: 10738: 10736: 10733: 10731: 10728: 10726: 10723: 10721: 10718: 10716: 10715:Factorization 10713: 10712: 10710: 10706: 10700: 10697: 10695: 10692: 10690: 10687: 10685: 10682: 10680: 10677: 10675: 10672: 10670: 10667: 10665: 10662: 10660: 10657: 10655: 10652: 10651: 10649: 10647:By properties 10645: 10639: 10636: 10634: 10631: 10629: 10626: 10622: 10619: 10618: 10617: 10614: 10610: 10607: 10606: 10605: 10602: 10598: 10595: 10594: 10593: 10590: 10586: 10583: 10582: 10581: 10578: 10576: 10573: 10571: 10568: 10567: 10565: 10563: 10558: 10554: 10550: 10543: 10538: 10536: 10531: 10529: 10524: 10523: 10520: 10512: 10508: 10504: 10501: 10497: 10496: 10491: 10486: 10485: 10474: 10470: 10466: 10461: 10457: 10453: 10449: 10444: 10440: 10434: 10430: 10426: 10421: 10417: 10411: 10407: 10403: 10398: 10394: 10388: 10384: 10380: 10376: 10372: 10367: 10363: 10357: 10352: 10351: 10345: 10344:"Polynomials" 10340: 10336: 10330: 10326: 10325: 10319: 10316: 10312: 10307: 10304: 10300: 10296: 10292: 10287: 10283: 10279: 10275: 10271: 10267: 10263: 10258: 10254: 10252:9789622092716 10248: 10244: 10243: 10237: 10233: 10229: 10225: 10219: 10215: 10211: 10207: 10203: 10198: 10192: 10188: 10184: 10179: 10176: 10174:0-201-01984-1 10170: 10166: 10162: 10157: 10153: 10147: 10143: 10142: 10136: 10133: 10131:0-534-93219-3 10127: 10123: 10119: 10114: 10110: 10104: 10100: 10099: 10093: 10090: 10088:0-395-14017-X 10084: 10080: 10076: 10071: 10067: 10061: 10057: 10056: 10050: 10049: 10036: 10030: 10022: 10016: 10012: 10006: 10000: 9993: 9989: 9983: 9976: 9970: 9962: 9958: 9954: 9947: 9943: 9934: 9932:0-03-029558-0 9928: 9924: 9917: 9909: 9907:9789491216503 9903: 9899: 9898: 9890: 9883: 9878: 9870: 9864: 9860: 9859: 9851: 9843: 9837: 9833: 9832: 9824: 9818:, p. 36. 9817: 9812: 9804: 9800: 9796: 9790: 9786: 9782: 9775: 9767: 9761: 9757: 9753: 9749: 9743: 9735: 9729: 9725: 9724: 9716: 9708: 9706:9789622092716 9702: 9698: 9697: 9689: 9681: 9675: 9671: 9667: 9663: 9656: 9649: 9645: 9640: 9633: 9629: 9624: 9617: 9613: 9608: 9594: 9590: 9583: 9575: 9569: 9565: 9558: 9551: 9546: 9544: 9536: 9532: 9526: 9522: 9521: 9513: 9505: 9501: 9500: 9492: 9485: 9481: 9475: 9471: 9470: 9462: 9455: 9451: 9450: 9443: 9441: 9432: 9426: 9422: 9421: 9413: 9405: 9399: 9395: 9394: 9386: 9384: 9382: 9375: 9371: 9366: 9357: 9356: 9351: 9348: 9341: 9334: 9330: 9325: 9323: 9321: 9306: 9305:brilliant.org 9302: 9296: 9294: 9279: 9275: 9268: 9266: 9259: 9253: 9247:, p. 82) 9246: 9241: 9234: 9229: 9222: 9217: 9211:, p. 96) 9210: 9205: 9198: 9193: 9189: 9180: 9177: 9176: 9170: 9167: 9161: 9156: 9152: 9148: 9144: 9140: 9139: 9134: 9128: 9118: 9114: 9110: 9106: 9093: 9086: 9082: 9078: 9068: 9066: 9062: 9061: 9056: 9051: 9049: 9045: 9041: 9037: 9033: 9029: 9019: 9017: 9013: 9009: 9006: 9003:defined on a 9002: 8998: 8994: 8990: 8986: 8980: 8976: 8972: 8968: 8958: 8954: 8950: 8944: 8939: 8935: 8928: 8923: 8919: 8913: 8911: 8907: 8891: 8886: 8882: 8878: 8875: 8870: 8866: 8862: 8859: 8856: 8851: 8848: 8845: 8841: 8835: 8832: 8829: 8825: 8821: 8816: 8812: 8806: 8802: 8798: 8795: 8787: 8785: 8781: 8769: 8765: 8759: 8744: 8742: 8738: 8734: 8733: 8727: 8723: 8721: 8715: 8711: 8707: 8706: 8701: 8698:Analogously, 8696: 8694: 8689: 8684: 8680: 8678: 8672: 8666: 8660: 8654: 8648: 8642: 8625: 8622: 8619: 8615: 8612: 8609: 8600: 8594: 8588: 8581: 8575: 8569: 8563: 8558: 8553: 8547: 8545: 8525: 8522: 8519: 8508:is a root of 8491: 8488: 8485: 8482: 8473: 8469: 8466: 8460: 8454: 8448: 8442: 8436: 8432: 8428: 8422: 8416: 8410: 8405: 8400: 8393: 8389: 8379: 8376: 8371: 8367: 8362: 8356: 8350: 8345: 8340: 8334: 8328: 8324:of the value 8323: 8317: 8313: 8307: 8302: 8299: 8294: 8288: 8284: 8279: 8273: 8267: 8261: 8259: 8254: 8249: 8245: 8244:finite fields 8239: 8233: 8228: 8223: 8220: 8214: 8210: 8204: 8198: 8193: 8189: 8184: 8178: 8172: 8169: 8164: 8159: 8153: 8147: 8142: 8137: 8131: 8125: 8121:The map from 8119: 8105: 8097: 8093: 8085: 8076: 8073: 8070: 8066: 8062: 8059: 8056: 8051: 8047: 8040: 8036: 8032: 8024: 8020: 8016: 8013: 8010: 8005: 8001: 7994: 7985: 7966: 7962: 7958: 7955: 7952: 7947: 7943: 7936: 7913: 7907: 7899: 7894: 7888: 7882: 7878: 7873: 7869: 7863: 7853: 7851: 7847: 7842: 7837: 7827: 7825: 7820: 7810: 7807: 7804: 7802: 7798: 7794: 7790: 7786: 7780: 7765: 7763: 7757: 7753: 7749: 7738: 7736: 7732: 7726: 7722: 7718: 7714: 7709: 7707: 7703: 7687: 7682: 7678: 7672: 7668: 7664: 7661: 7658: 7653: 7649: 7643: 7639: 7635: 7632: 7627: 7623: 7619: 7616: 7611: 7607: 7603: 7597: 7593: 7587: 7583: 7576: 7571: 7568: 7565: 7561: 7557: 7551: 7545: 7537: 7521: 7516: 7512: 7506: 7502: 7498: 7495: 7492: 7487: 7483: 7477: 7473: 7469: 7466: 7461: 7457: 7453: 7448: 7444: 7440: 7434: 7430: 7424: 7420: 7413: 7408: 7405: 7402: 7398: 7394: 7388: 7382: 7374: 7370: 7364: 7354: 7352: 7348: 7344: 7343:interpolation 7340: 7335: 7333: 7329: 7324: 7322: 7318: 7314: 7310: 7306: 7302: 7298: 7294: 7290: 7286: 7282: 7278: 7273: 7271: 7267: 7263: 7259: 7255: 7251: 7247: 7241: 7231: 7223: 7221: 7217: 7213: 7209: 7205: 7200: 7198: 7194: 7190: 7185: 7183: 7179: 7175: 7171: 7167: 7162: 7160: 7159: 7154: 7150: 7146: 7142: 7137: 7135: 7131: 7127: 7123: 7119: 7118:Galois theory 7114: 7110: 7106: 7102: 7098: 7093: 7089: 7073: 7070: 7067: 7064: 7061: 7058: 7053: 7049: 7028: 7024: 7015: 7010: 7007: 6997: 6993: 6983: 6977: 6972: 6969: 6965: 6960: 6956: 6952: 6946: 6940: 6938: 6934: 6930: 6925: 6919: 6913: 6909:as a root of 6907: 6902: 6897: 6890: 6886: 6879: 6873: 6867: 6862: 6857: 6851: 6846: 6845:multiple root 6841: 6835: 6829: 6825: 6814: 6810: 6806: 6800: 6794: 6788: 6784: 6780: 6775: 6769: 6763: 6760: 6755: 6749: 6745: 6741: 6731: 6727: 6713: 6708: 6702: 6698: 6692: 6682: 6680: 6676: 6672: 6667: 6665: 6661: 6657: 6653: 6649: 6645: 6641: 6636: 6633: 6629: 6625: 6621: 6617: 6613: 6607: 6606: 6600: 6596: 6591: 6577: 6574: 6571: 6568: 6565: 6562: 6559: 6554: 6550: 6546: 6539:For example, 6526: 6523: 6518: 6514: 6510: 6507: 6502: 6498: 6494: 6489: 6485: 6479: 6475: 6471: 6468: 6465: 6460: 6457: 6454: 6450: 6444: 6441: 6438: 6434: 6430: 6425: 6421: 6415: 6411: 6402: 6398: 6397: 6392: 6386: 6376: 6373: 6371: 6367: 6363: 6360:. It has two 6359: 6355: 6351: 6340: 6335: 6331: 6325: 6321: 6317: 6313: 6307: 6303: 6297: 6290: 6286: 6282: 6276: 6273: 6263: 6257: 6251: 6247: 6241: 6237: 6231: 6224: 6220: 6216: 6210: 6207: 6197: 6191: 6185: 6181: 6175: 6168: 6164: 6160: 6154: 6147: 6143: 6134: 6128: 6115: 6109: 6103: 6096: 6092: 6088: 6082: 6075: 6069: 6056: 6047: 6043: 6039: 6033: 6029: 6021: 6017: 6011: 6010: 6009: 6007: 5995: 5987: 5983: 5979: 5975: 5971: 5967: 5963: 5959: 5949: 5944: 5938: 5934: 5927: 5923: 5919: 5915: 5911: 5907: 5897: 5892: 5886: 5880: 5876: 5872: 5868: 5864: 5860: 5850: 5845: 5837: 5833: 5829: 5825: 5821: 5817: 5807: 5802: 5796: 5792: 5788: 5780: 5776: 5772: 5768: 5764: 5754: 5749: 5743: 5739: 5731: 5727: 5723: 5719: 5709: 5704: 5698: 5694: 5690: 5680: 5675: 5669: 5665: 5655: 5650: 5649: 5642: 5629: 5624: 5620: 5616: 5613: 5605: 5601: 5597: 5594: 5586: 5582: 5578: 5575: 5567: 5563: 5559: 5556: 5553: 5550: 5542: 5539: 5536: 5532: 5528: 5525: 5517: 5514: 5511: 5507: 5503: 5500: 5495: 5491: 5467: 5462: 5459: 5458: 5452: 5450: 5446: 5442: 5437: 5420: 5417: 5414: 5411: 5386: 5382: 5378: 5375: 5355: 5350: 5345: 5338: 5334: 5330: 5327: 5322: 5300: 5297: 5292: 5288: 5284: 5279: 5275: 5271: 5268: 5265: 5260: 5256: 5252: 5249: 5244: 5240: 5236: 5233: 5227: 5224: 5221: 5215: 5195: 5192: 5189: 5184: 5180: 5176: 5170: 5164: 5157:, defined by 5155: 5149: 5147: 5146:real function 5143: 5138: 5133: 5123: 5116: 5109: 5103: 5094: 5071: 5065: 5043: 5039: 5035: 5032: 5027: 5023: 5019: 5014: 5010: 5004: 5000: 4996: 4993: 4990: 4985: 4982: 4979: 4975: 4969: 4966: 4963: 4959: 4955: 4950: 4946: 4940: 4936: 4927: 4922: 4917: 4913: 4907: 4897: 4890: 4886: 4880: 4873: 4869: 4858: 4854: 4848: 4844: 4840: 4835: 4832: 4828: 4824: 4802: 4778: 4775: 4772: 4765: 4762: 4759: 4755: 4749: 4745: 4736: 4731: 4728: 4725: 4721: 4717: 4714: 4711: 4708: 4705: 4702: 4697: 4693: 4689: 4684: 4678: 4674: 4668: 4664: 4657: 4652: 4646: 4642: 4636: 4632: 4625: 4622: 4619: 4614: 4608: 4604: 4598: 4595: 4592: 4588: 4581: 4575: 4572: 4569: 4562: 4559: 4556: 4552: 4546: 4542: 4518: 4510: 4494: 4489: 4486: 4483: 4479: 4473: 4469: 4465: 4460: 4455: 4452: 4449: 4445: 4441: 4436: 4432: 4428: 4425: 4420: 4416: 4412: 4409: 4406: 4403: 4398: 4395: 4392: 4388: 4382: 4379: 4376: 4372: 4365: 4362: 4359: 4353: 4348: 4345: 4342: 4338: 4332: 4328: 4324: 4298: 4294: 4288: 4284: 4278: 4273: 4270: 4267: 4263: 4259: 4254: 4250: 4246: 4243: 4238: 4234: 4230: 4225: 4221: 4215: 4211: 4207: 4204: 4201: 4196: 4193: 4190: 4186: 4180: 4177: 4174: 4170: 4166: 4161: 4157: 4151: 4147: 4143: 4140: 4132: 4128: 4122: 4112: 4110: 4106: 4103: 4099: 4098:factorization 4094: 4079: 4073: 4067: 4062: 4059: 4056: 4050: 4047: 4043: 4038: 4032: 4026: 4021: 4018: 4015: 4009: 4006: 4002: 3995: 3992: 3989: 3983: 3962: 3958: 3955: 3952: 3949: 3944: 3940: 3935: 3928: 3925: 3922: 3916: 3896: 3893: 3888: 3884: 3880: 3872: 3868: 3864: 3860: 3856: 3852: 3842: 3840: 3834: 3830: 3826: 3820: 3816: 3809: 3805: 3800: 3791: 3787: 3783: 3779: 3774: 3768: 3764: 3758: 3756: 3752: 3746: 3742: 3735: 3731: 3728: 3724: 3717: 3713: 3709: 3703: 3699: 3695: 3689: 3685: 3681: 3677: 3672: 3668: 3663: 3655: 3649: 3645: 3641: 3640: 3635: 3631: 3630: 3619: 3605: 3599: 3596: 3593: 3590: 3584: 3581: 3576: 3568: 3565: 3562: 3559: 3553: 3544: 3538: 3532: 3529: 3523: 3514: 3511: 3508: 3485: 3482: 3479: 3476: 3473: 3467: 3461: 3441: 3438: 3435: 3430: 3426: 3422: 3416: 3410: 3390: 3387: 3384: 3377: 3357: 3343: 3329: 3326: 3323: 3320: 3317: 3312: 3308: 3304: 3301: 3298: 3293: 3289: 3285: 3282: 3279: 3276: 3273: 3270: 3265: 3261: 3257: 3254: 3251: 3248: 3245: 3242: 3237: 3233: 3229: 3226: 3223: 3220: 3196: 3191: 3183: 3180: 3177: 3174: 3171: 3163: 3156: 3152: 3148: 3145: 3140: 3133: 3129: 3125: 3120: 3108: 3105: 3102: 3099: 3096: 3088: 3083: 3078: 3074: 3070: 3065: 3057: 3054: 3051: 3048: 3045: 3042: 3039: 3036: 3033: 3030: 3027: 3019: 3012: 3008: 3004: 2998: 2993: 2990: 2962: 2957: 2952: 2949: 2946: 2941: 2936: 2933: 2928: 2923: 2920: 2915: 2906: 2903: 2898: 2891: 2887: 2883: 2880: 2875: 2868: 2864: 2860: 2855: 2850: 2847: 2844: 2839: 2830: 2827: 2822: 2817: 2812: 2808: 2804: 2799: 2794: 2791: 2788: 2783: 2776: 2772: 2768: 2762: 2757: 2754: 2720: 2714: 2708: 2697: 2686: 2683: 2677: 2671: 2660: 2649: 2646: 2640: 2634: 2623: 2612: 2609: 2603: 2597: 2586: 2571: 2565: 2559: 2556: 2545: 2534: 2531: 2525: 2519: 2516: 2505: 2494: 2491: 2485: 2479: 2476: 2465: 2454: 2451: 2445: 2439: 2436: 2425: 2410: 2404: 2398: 2395: 2384: 2373: 2370: 2364: 2358: 2355: 2344: 2333: 2330: 2324: 2318: 2315: 2304: 2293: 2290: 2284: 2278: 2275: 2262: 2253: 2244: 2211: 2208: 2205: 2202: 2199: 2196: 2193: 2190: 2187: 2184: 2181: 2173: 2163: 2160: 2157: 2154: 2151: 2148: 2145: 2142: 2134: 2121: 2115: 2105: 2102: 2088: 2085: 2080: 2076: 2072: 2069: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2019: 2016: 2013: 2007: 2002: 1998: 1994: 1991: 1988: 1985: 1982: 1979: 1973: 1970: 1967: 1964: 1961: 1958: 1952: 1944: 1940: 1936: 1933: 1928: 1924: 1920: 1914: 1911: 1908: 1905: 1885: 1882: 1877: 1873: 1869: 1866: 1863: 1860: 1857: 1852: 1848: 1844: 1841: 1838: 1835: 1832: 1829: 1826: 1823: 1820: 1817: 1814: 1809: 1805: 1801: 1798: 1795: 1792: 1789: 1782:then the sum 1769: 1766: 1761: 1757: 1753: 1750: 1747: 1744: 1741: 1736: 1732: 1728: 1725: 1722: 1719: 1699: 1696: 1693: 1690: 1687: 1684: 1681: 1678: 1675: 1670: 1666: 1662: 1659: 1656: 1648: 1644: 1629: 1626: 1620: 1616: 1611: 1607: 1603: 1599: 1595: 1590: 1588: 1584: 1580: 1576: 1572: 1568: 1564: 1560: 1556: 1551: 1549: 1545: 1541: 1537: 1532: 1530: 1514: 1504: 1497: 1493: 1486: 1480: 1475: 1470: 1468: 1463: 1459: 1456: 1452: 1449: 1442: 1435: 1429: 1422: 1417: 1415: 1409: 1405: 1400: 1396: 1391: 1387: 1383: 1377: 1371: 1367: 1363: 1359: 1358: 1354: 1350: 1346: 1341: 1327: 1317: 1288: 1284: 1277: 1262: 1233: 1229: 1226: 1219: 1204: 1175: 1169: 1165: 1161: 1155: 1141: 1134: 1128: 1122: 1116: 1095: 1090: 1086: 1082: 1079: 1072:For example: 1070: 1068: 1064: 1063:constant term 1059: 1056: 1052: 1044: 1034: 1032: 1028: 1010: 1006: 1000: 996: 990: 985: 982: 979: 975: 966: 961: 959: 955: 939: 919: 911: 893: 889: 885: 882: 879: 874: 870: 849: 844: 840: 836: 833: 828: 824: 820: 815: 811: 805: 801: 797: 794: 791: 786: 783: 780: 776: 770: 767: 764: 760: 756: 751: 747: 741: 737: 727: 721: 707: 704: 701: 698: 695: 690: 686: 682: 676: 673: 670: 661: 658: 655: 645: 629: 626: 623: 620: 617: 612: 608: 584: 581: 578: 569: 566: 563: 552: 548: 547:associativity 544: 543:commutativity 540: 536: 532: 528: 524: 520: 516: 512: 508: 504: 500: 496: 486: 472: 469: 466: 460: 454: 446: 442: 438: 434: 430: 426: 422: 417: 415: 411: 407: 403: 399: 395: 391: 388:which is the 375: 369: 363: 357: 349: 345: 341: 337: 333: 329: 325: 320: 318: 314: 310: 306: 302: 298: 295: 291: 287: 283: 279: 275: 271: 267: 263: 260:A polynomial 258: 256: 252: 248: 244: 243:indeterminate 240: 236: 228: 223: 214: 212: 208: 204: 200: 196: 195: 190: 186: 182: 179: 169: 167: 163: 159: 155: 151: 147: 143: 139: 135: 131: 127: 123: 122:word problems 119: 114: 110: 106: 102: 95: 91: 85: 80: 76: 72: 68: 64: 60: 56: 53:(also called 52: 48: 44: 40: 33: 19: 10745:Discriminant 10664:Multivariate 10548: 10511:the original 10493: 10490:"Polynomial" 10472: 10468: 10455: 10451: 10424: 10405: 10374: 10354:. Springer. 10349: 10327:. Springer. 10323: 10310: 10290: 10265: 10261: 10241: 10209: 10182: 10160: 10140: 10117: 10101:. Springer. 10097: 10074: 10058:. Springer. 10054: 10029: 10010: 9999: 9991: 9982: 9969: 9960: 9957:prime number 9946: 9922: 9916: 9900:. Springer. 9896: 9889: 9884:, p. 75 9877: 9857: 9850: 9830: 9823: 9811: 9780: 9774: 9751: 9742: 9726:. Elsevier. 9722: 9715: 9695: 9688: 9669: 9655: 9639: 9628:Barbeau 2003 9623: 9612:Barbeau 2003 9607: 9596:. Retrieved 9592: 9582: 9563: 9557: 9534: 9519: 9512: 9498: 9491: 9483: 9468: 9461: 9453: 9448: 9419: 9412: 9392: 9370:Edwards 1995 9365: 9353: 9340: 9329:Barbeau 2003 9308:. Retrieved 9304: 9281:. Retrieved 9277: 9274:"Polynomial" 9257: 9252: 9240: 9228: 9216: 9204: 9192: 9165: 9159: 9155:La géometrie 9154: 9146: 9136: 9130: 9112: 9108: 9104: 9088: 9058: 9052: 9025: 8982: 8952: 8948: 8942: 8937: 8933: 8926: 8921: 8917: 8914: 8909: 8905: 8788: 8783: 8779: 8761: 8747:Applications 8730: 8717: 8713: 8709: 8703: 8699: 8697: 8687: 8682: 8675: 8670: 8664: 8658: 8652: 8646: 8640: 8598: 8592: 8586: 8579: 8573: 8567: 8561: 8551: 8548: 8512:if and only 8471: 8467: 8464: 8458: 8452: 8446: 8440: 8434: 8430: 8426: 8420: 8414: 8408: 8398: 8395: 8382:Divisibility 8374: 8360: 8354: 8348: 8338: 8332: 8326: 8322:substitution 8315: 8311: 8305: 8292: 8286: 8282: 8277: 8271: 8265: 8262: 8252: 8248:prime number 8237: 8231: 8224: 8218: 8212: 8208: 8202: 8196: 8191: 8187: 8182: 8176: 8173: 8167: 8157: 8151: 8145: 8135: 8129: 8123: 8120: 7986: 7897: 7892: 7886: 7880: 7871: 7867: 7865: 7850:power series 7839: 7830:Power series 7822: 7808: 7805: 7782: 7755: 7751: 7747: 7744: 7734: 7727: 7720: 7716: 7712: 7710: 7701: 7535: 7366: 7336: 7325: 7320: 7316: 7308: 7304: 7296: 7292: 7288: 7284: 7280: 7276: 7274: 7265: 7261: 7257: 7248:is a finite 7245: 7243: 7229: 7206:is called a 7201: 7186: 7165: 7163: 7156: 7138: 7122:group theory 7111:). In 1830, 6996:golden ratio 6981: 6973: 6967: 6963: 6951:real numbers 6944: 6941: 6923: 6917: 6911: 6905: 6901:multiplicity 6900: 6895: 6888: 6884: 6877: 6871: 6865: 6860: 6855: 6849: 6844: 6839: 6833: 6827: 6823: 6812: 6808: 6804: 6798: 6792: 6786: 6782: 6773: 6767: 6764: 6758: 6747: 6743: 6729: 6725: 6711: 6706: 6704: 6675:multiplicity 6668: 6637: 6631: 6627: 6623: 6619: 6615: 6611: 6602: 6598: 6592: 6403:of the form 6394: 6390: 6388: 6374: 6369: 6365: 6347: 6338: 6333: 6329: 6323: 6319: 6315: 6311: 6305: 6301: 6295: 6288: 6284: 6280: 6261: 6255: 6249: 6245: 6239: 6235: 6229: 6222: 6218: 6214: 6195: 6189: 6183: 6179: 6173: 6166: 6162: 6158: 6145: 6132: 6126: 6113: 6107: 6101: 6094: 6090: 6086: 6073: 6067: 6054: 6045: 6041: 6037: 6027: 6019: 6015: 6003: 5993: 5985: 5981: 5977: 5973: 5969: 5965: 5961: 5957: 5936: 5932: 5925: 5921: 5917: 5913: 5909: 5905: 5884: 5878: 5874: 5870: 5866: 5862: 5858: 5835: 5831: 5827: 5823: 5819: 5815: 5794: 5790: 5786: 5778: 5774: 5770: 5766: 5762: 5741: 5737: 5729: 5725: 5721: 5717: 5696: 5692: 5688: 5667: 5663: 5463: 5456: 5453: 5438: 5153: 5150: 5128: 5121: 5114: 5107: 5101: 4920: 4911: 4909: 4888: 4884: 4878: 4871: 4867: 4856: 4852: 4846: 4843:prime number 4836: 4830: 4826: 4822: 4800: 4125:Calculating 4124: 4097: 4095: 3867:real numbers 3848: 3832: 3828: 3818: 3814: 3807: 3803: 3789: 3785: 3781: 3777: 3766: 3762: 3759: 3744: 3740: 3733: 3729: 3726: 3722: 3720:, such that 3715: 3711: 3707: 3701: 3697: 3693: 3687: 3683: 3679: 3675: 3664: 3653: 3637: 3633: 3627: 3625: 3349: 2117: 2103: 1640: 1624: 1618: 1614: 1609: 1605: 1601: 1597: 1593: 1591: 1582: 1574: 1570: 1554: 1552: 1533: 1528: 1512: 1502: 1495: 1491: 1484: 1478: 1471: 1461: 1457: 1454: 1450: 1447: 1440: 1433: 1427: 1420: 1418: 1413: 1407: 1403: 1394: 1392: 1385: 1381: 1375: 1369: 1365: 1361: 1355: 1352: 1348: 1344: 1342: 1142: 1132: 1126: 1120: 1114: 1071: 1066: 1060: 1054: 1050: 1046: 962: 957: 910:coefficients 909: 725: 722: 538: 513:by means of 510: 506: 494: 492: 444: 440: 436: 432: 424: 420: 418: 413: 409: 405: 397: 393: 389: 347: 343: 339: 335: 331: 327: 323: 321: 316: 312: 308: 304: 300: 296: 289: 285: 281: 277: 273: 269: 265: 261: 259: 254: 246: 242: 238: 234: 232: 210: 209:). The word 202: 198: 192: 188: 184: 183:: the Greek 177: 175: 125: 115: 108: 104: 100: 93: 89: 83: 59:coefficients 42: 36: 10771:Polynomials 10694:Homogeneous 10689:Square-free 10684:Irreducible 10549:Polynomials 10324:Polynomials 10313:, Reading: 10268:: 280–313. 10206:Lang, Serge 10055:Polynomials 9630:, pp. 9614:, pp. 9331:, pp. 9245:Moise (1967 9233:McCoy (1968 9057:the phrase 9032:eigenvalues 8143:, by which 7725:matrix ring 6861:simple root 6715:is a value 6603:polynomial 6272:cubic curve 6130:-intercept 6071:-intercept 5912:) = 1/100 ( 4127:derivatives 3797:, then the 3376:composition 3346:Composition 1421:homogeneous 1031:coefficient 956:, called a 427:, then the 67:subtraction 39:mathematics 10765:Categories 10654:Univariate 10293:, Boston: 10077:, Boston: 10045:References 9882:McCoy 1968 9803:1170.15300 9646:, p. 9598:2020-07-25 9372:, p. 9310:2020-08-28 9283:2020-08-28 8965:See also: 8462:such that 7868:polynomial 7319:) and cos( 7307:) and cos( 7295:) and cos( 7287:) and cos( 7279:) and cos( 7260:) and cos( 7174:algorithms 7149:algorithms 6881:such that 6802:such that 6723:such that 6695:See also: 6597:, and the 5865:) = 1/20 ( 5822:) = 1/14 ( 5457:evaluation 5441:continuous 5142:restricted 4916:evaluating 4904:See also: 4865:copies of 4131:derivative 4105:algorithms 3825:evaluation 1632:Operations 1610:trivariate 539:polynomial 499:expression 489:Definition 211:polynomial 178:polynomial 43:polynomial 10740:Resultant 10679:Trinomial 10659:Bivariate 10500:EMS Press 10282:197662587 9355:MathWorld 9065:algorithm 8860:⋯ 8849:− 8833:− 8523:− 8486:∈ 8206:(that is 8074:− 8060:… 8014:… 7987:One has 7956:… 7662:⋯ 7562:∑ 7496:⋯ 7399:∑ 7299:) (using 7254:functions 7212:algorithm 7065:− 7059:− 6765:A number 6599:solutions 6569:− 6469:⋯ 6458:− 6442:− 6379:Equations 6358:asymptote 5557:⋯ 5540:− 5515:− 5412:− 5379:− 5331:− 5298:− 5190:− 4994:⋯ 4983:− 4967:− 4722:∑ 4623:⋯ 4596:− 4487:− 4446:∑ 4407:⋯ 4396:− 4380:− 4363:− 4346:− 4264:∑ 4205:⋯ 4194:− 4178:− 4060:− 3993:− 3926:− 3894:− 3845:Factoring 3708:remainder 3512:∘ 3388:∘ 2715:⋅ 2678:⋅ 2641:⋅ 2604:⋅ 2566:⋅ 2526:⋅ 2486:⋅ 2446:⋅ 2405:⋅ 2365:⋅ 2325:⋅ 2285:⋅ 2017:− 1959:− 1934:− 1842:− 1836:− 1815:− 1726:− 1697:− 1676:− 1606:bivariate 1548:trinomial 1412:, is the 1289:⏟ 1234:⏟ 1220:− 1176:⏟ 1138:2 + 1 = 3 1080:− 976:∑ 883:… 795:⋯ 784:− 768:− 696:− 674:− 659:− 618:− 582:− 567:− 507:variables 503:constants 361:↦ 207:monomials 176:The word 172:Etymology 138:economics 130:chemistry 55:variables 10725:Division 10674:Binomial 10669:Monomial 10475:: 245–8. 10458:: 245–8. 10425:Calculus 10303:68015225 10208:(2002), 9750:(1981). 9504:OpenStax 9173:See also 9149:, 1544. 9008:interval 8985:calculus 8975:B-spline 8538:divides 8366:analysis 8133:sending 7789:quotient 7204:integers 7153:computer 6893:divides 6831:divides 6790:divides 6605:identity 6595:unknowns 6401:equation 6399:, is an 6337:≠ 0 and 6327:, where 6259:, where 6206:parabola 6193:, where 6111:, where 6052:, where 5887:− 3) + 2 5087:for all 4926:argument 4115:Calculus 3694:quotient 3644:integers 3622:Division 1563:function 1544:binomial 1540:monomial 1349:constant 954:function 644:equality 515:addition 251:function 239:variable 194:binomial 146:calculus 63:addition 10776:Algebra 10232:1878556 10210:Algebra 9994:degree. 9668:(ed.). 9552:, §5.4] 9506:. §7.1. 9071:History 9016:splines 9005:compact 8916:0 < 8431:divides 8163:algebra 7875:over a 7787:is the 7704:is the 7275:If sin( 7264:) with 7178:complex 7126:algebra 6984:− 1 = 0 6929:complex 6671:complex 6640:algebra 6031:-axis. 6025:is the 5785:= 1/4 ( 5137:complex 5127:, ..., 5099:(here, 5091:in the 4924:of one 4820:⁠ 4808:⁠ 3823:is the 3739:degree( 2120:product 1587:complex 1579:integer 1438:degree 1425:degree 1416:-axis. 531:numbers 162:algebra 134:physics 10562:degree 10435: 10412: 10389: 10358: 10331: 10301: 10280: 10249: 10230: 10220: 10193: 10171: 10148: 10128: 10105: 10085: 10062: 10017: 9953:modulo 9929: 9904: 9865: 9838: 9801: 9791: 9762: 9730: 9703: 9676: 9570: 9527: 9476: 9427: 9400: 9083:, and 9038:of an 9034:. The 8977:, and 8904:where 8402:is an 8298:unital 8227:ideals 8161:is an 7700:where 7090:, the 6314:+ ⋯ + 5781:/2 − 2 5777:/4 − 3 5773:/4 + 3 5645:Graphs 5449:entire 5447:, and 5445:smooth 5093:domain 4839:modulo 4794:where 3706:and a 3636:, or 1622:, and 1567:domain 1565:, the 1529:degree 1065:and a 862:where 497:is an 284:, not 272:or as 241:or an 57:) and 10278:S2CID 10035:field 10005:monic 9992:every 9955:some 9664:. In 9185:Notes 9153:, in 9048:graph 9046:of a 8941:< 8925:< 8768:radix 8691:is a 8602:with 8577:with 8557:field 8555:is a 8504:then 8475:. If 8303:over 8256:(see 8165:over 7166:zeros 6869:. If 6859:is a 6843:is a 6821:) of 6752:or a 6750:) = 0 6732:) = 0 6648:cubic 6608:like 6270:is a 6204:is a 6142:slope 6022:) = 0 6006:graph 5984:+ 1)( 5976:− 1)( 5972:− 2)( 5968:− 3)( 5964:) = ( 5939:− 80) 5931:+ 145 5877:+ 1)( 5873:+ 2)( 5869:+ 4)( 5840:+ 0.5 5838:− 3) 5834:− 1)( 5830:+ 1)( 5826:+ 4)( 5793:+ 1)( 5789:+ 4)( 5740:+ 1)( 5695:) = 2 5670:) = 2 4841:some 3855:field 3773:monic 3646:is a 3498:then 2230:then 1410:) = 0 1399:roots 1027:terms 525:to a 429:image 227:graph 203:poly- 189:nomen 45:is a 10551:and 10473:1892 10456:1884 10433:ISBN 10410:ISBN 10387:ISBN 10356:ISBN 10329:ISBN 10299:LCCN 10247:ISBN 10218:ISBN 10191:ISBN 10169:ISBN 10146:ISBN 10126:ISBN 10103:ISBN 10083:ISBN 10060:ISBN 10015:ISBN 9927:ISBN 9902:ISBN 9863:ISBN 9836:ISBN 9789:ISBN 9760:ISBN 9728:ISBN 9701:ISBN 9674:ISBN 9568:ISBN 9525:ISBN 9474:ISBN 9425:ISBN 9398:ISBN 9115:= 29 8946:for 8932:0 ≤ 8930:and 8720:unit 8668:and 8656:and 8590:and 8565:and 8559:and 8412:and 8406:and 8390:and 7326:For 7256:sin( 7132:and 7120:and 7099:and 6754:zero 6707:root 6699:and 6650:and 6626:) = 6287:) = 6221:) = 6165:) = 6140:and 6093:) = 6044:) = 5996:+ 3) 5988:+ 2) 5935:− 26 5924:+ 28 5920:− 26 5881:− 1) 5797:− 2) 5769:) = 5744:− 2) 5724:) = 5454:The 3784:) = 3753:and 3737:and 3656:+ 1) 3454:and 1712:and 1559:real 1472:The 1360:and 1118:and 600:and 549:and 521:and 443:for 402:ring 338:for 233:The 225:The 185:poly 164:and 156:and 148:and 140:and 132:and 73:and 41:, a 10560:By 10379:doi 10270:doi 9799:Zbl 9145:'s 9135:'s 9107:+ 2 8987:is 8955:− 1 8681:or 8596:in 8582:≠ 0 8549:If 8456:in 8438:or 8396:If 8336:in 8320:by 8275:in 8263:If 8260:). 8240:+ 1 8190:to 8127:to 7737:). 7538:is 7345:of 7252:of 7161:). 7136:). 6988:1/2 6986:is 6947:+ 1 6903:of 6863:of 6847:of 6815:) Q 6807:= ( 6719:of 6372:). 6341:≥ 2 6267:≠ 0 6201:≠ 0 6119:≠ 0 6060:≠ 0 5916:− 2 5736:= ( 5732:− 2 5699:+ 1 5095:of 4803:+ 1 4531:is 3909:is 3812:by 3771:is 3652:1/( 1522:is 1498:+ 4 1494:− 5 1460:− 3 1453:+ 7 1434:all 1432:if 1423:of 1388:+ 1 1384:+ 2 1379:in 509:or 431:of 342:in 199:bi- 136:to 111:+ 1 105:xyz 103:+ 2 96:+ 7 92:− 4 87:is 77:to 37:In 10767:: 10498:, 10492:, 10471:. 10467:. 10454:. 10450:. 10431:. 10385:. 10373:. 10346:. 10297:, 10276:. 10266:45 10264:. 10228:MR 10226:, 10212:, 10189:. 10185:. 10167:, 10124:, 10081:, 9797:. 9787:. 9758:. 9754:. 9648:38 9634:–5 9632:64 9618:–2 9616:80 9591:. 9542:^ 9533:. 9502:. 9482:. 9439:^ 9380:^ 9374:78 9352:. 9335:–2 9319:^ 9303:. 9292:^ 9276:. 9264:^ 9117:. 9111:+ 9096:c. 9094:, 9079:, 9018:. 8973:, 8969:, 8957:. 8910:r' 8739:. 8695:. 8546:. 8470:= 8378:. 8281:a 8213:rx 8211:= 8209:xr 8171:. 7866:A 7803:. 7783:A 7764:. 7754:, 7711:A 7708:. 7367:A 7353:. 7334:. 7321:nx 7317:nx 7281:nx 7277:nx 7262:nx 7258:nx 7244:A 7222:. 7199:. 7184:. 7074:0. 6966:− 6939:. 6887:− 6826:− 6811:− 6785:− 6705:A 6681:. 6630:− 6622:− 6618:)( 6614:+ 6527:0. 6389:A 6304:+ 6294:+ 6274:. 6248:+ 6238:+ 6228:+ 6208:. 6182:+ 6172:+ 6152:. 6100:+ 6008:. 5980:)( 5728:− 5451:. 5443:, 5301:7. 5120:, 5113:, 4910:A 4896:. 4887:+ 4853:ka 4834:. 4829:+ 4825:+ 4111:. 3788:− 3757:. 3732:+ 3725:= 3682:)/ 3662:. 3632:, 3330:5. 3321:28 3286:15 3277:12 3246:21 3181:25 3126:15 3106:10 3028:10 2963:5. 2934:25 2921:10 2861:15 2789:10 2089:6. 1617:, 1608:, 1553:A 1550:. 1524:−5 1469:. 1140:. 1110:−5 1053:= 967:: 960:. 720:. 545:, 517:, 493:A 168:. 113:. 109:yz 107:− 69:, 65:, 10541:e 10534:t 10527:v 10441:. 10418:. 10395:. 10381:: 10364:. 10337:. 10284:. 10272:: 10255:. 10201:. 10199:. 10154:. 10111:. 10068:. 10037:. 10023:. 9977:. 9964:. 9961:p 9935:. 9910:. 9871:. 9844:. 9805:. 9768:. 9736:. 9709:. 9682:. 9650:. 9601:. 9576:. 9433:. 9406:. 9358:. 9333:1 9313:. 9286:. 9166:x 9160:a 9113:z 9109:y 9105:x 9103:3 8953:m 8949:i 8943:b 8938:i 8934:r 8927:b 8922:m 8918:r 8906:m 8892:, 8887:0 8883:r 8879:+ 8876:b 8871:1 8867:r 8863:+ 8857:+ 8852:1 8846:m 8842:b 8836:1 8830:m 8826:r 8822:+ 8817:m 8813:b 8807:m 8803:r 8799:= 8796:a 8784:a 8780:b 8722:" 8688:F 8671:g 8665:f 8659:r 8653:q 8647:g 8641:r 8626:r 8623:+ 8620:g 8616:q 8613:= 8610:f 8599:F 8593:r 8587:q 8580:g 8574:F 8568:g 8562:f 8552:F 8540:f 8526:a 8520:x 8510:f 8506:a 8492:, 8489:R 8483:a 8472:g 8468:q 8465:f 8459:R 8453:q 8447:g 8441:f 8435:g 8427:f 8421:R 8415:g 8409:f 8399:R 8375:x 8361:R 8355:p 8349:R 8339:P 8333:x 8327:r 8318:) 8316:r 8314:( 8312:f 8306:R 8293:R 8287:f 8278:R 8272:P 8266:R 8253:R 8238:x 8232:R 8219:x 8203:R 8197:x 8192:R 8188:x 8183:R 8177:R 8168:R 8158:R 8152:R 8146:R 8136:r 8130:R 8124:R 8106:. 8103:] 8098:n 8094:x 8090:[ 8086:) 8082:] 8077:1 8071:n 8067:x 8063:, 8057:, 8052:1 8048:x 8044:[ 8041:R 8037:( 8033:= 8030:] 8025:n 8021:x 8017:, 8011:, 8006:1 8002:x 7998:[ 7995:R 7972:] 7967:n 7963:x 7959:, 7953:, 7948:1 7944:x 7940:[ 7937:R 7917:] 7914:x 7911:[ 7908:R 7893:R 7887:R 7881:R 7872:f 7791:( 7758:) 7756:e 7752:x 7750:( 7748:P 7735:R 7733:( 7730:n 7728:M 7721:A 7702:I 7688:, 7683:n 7679:A 7673:n 7669:a 7665:+ 7659:+ 7654:2 7650:A 7644:2 7640:a 7636:+ 7633:A 7628:1 7624:a 7620:+ 7617:I 7612:0 7608:a 7604:= 7598:i 7594:A 7588:i 7584:a 7577:n 7572:0 7569:= 7566:i 7558:= 7555:) 7552:A 7549:( 7546:P 7536:A 7522:, 7517:n 7513:x 7507:n 7503:a 7499:+ 7493:+ 7488:2 7484:x 7478:2 7474:a 7470:+ 7467:x 7462:1 7458:a 7454:+ 7449:0 7445:a 7441:= 7435:i 7431:x 7425:i 7421:a 7414:n 7409:0 7406:= 7403:i 7395:= 7392:) 7389:x 7386:( 7383:P 7309:x 7305:x 7297:x 7293:x 7289:x 7285:x 7266:n 7071:= 7068:1 7062:x 7054:2 7050:x 7029:2 7025:/ 7021:) 7016:5 7011:+ 7008:1 7005:( 6982:x 6980:2 6968:a 6964:x 6945:x 6924:P 6918:P 6912:P 6906:a 6896:P 6891:) 6889:a 6885:x 6883:( 6878:m 6872:P 6866:P 6856:a 6850:P 6840:a 6834:P 6828:a 6824:x 6819:1 6813:a 6809:x 6805:P 6799:Q 6793:P 6787:a 6783:x 6774:P 6768:a 6759:P 6748:x 6746:( 6744:P 6736:P 6730:a 6728:( 6726:P 6721:x 6717:a 6712:P 6632:y 6628:x 6624:y 6620:x 6616:y 6612:x 6610:( 6578:0 6575:= 6572:5 6566:x 6563:4 6560:+ 6555:2 6551:x 6547:3 6524:= 6519:0 6515:a 6511:+ 6508:x 6503:1 6499:a 6495:+ 6490:2 6486:x 6480:2 6476:a 6472:+ 6466:+ 6461:1 6455:n 6451:x 6445:1 6439:n 6435:a 6431:+ 6426:n 6422:x 6416:n 6412:a 6370:x 6366:x 6339:n 6334:n 6330:a 6324:x 6320:n 6316:a 6312:x 6309:2 6306:a 6302:x 6299:1 6296:a 6292:0 6289:a 6285:x 6283:( 6281:f 6265:3 6262:a 6256:x 6253:3 6250:a 6246:x 6243:2 6240:a 6236:x 6233:1 6230:a 6226:0 6223:a 6219:x 6217:( 6215:f 6199:2 6196:a 6190:x 6187:2 6184:a 6180:x 6177:1 6174:a 6170:0 6167:a 6163:x 6161:( 6159:f 6149:1 6146:a 6136:0 6133:a 6127:y 6121:, 6117:1 6114:a 6108:x 6105:1 6102:a 6098:0 6095:a 6091:x 6089:( 6087:f 6077:0 6074:a 6068:y 6062:, 6058:0 6055:a 6049:0 6046:a 6042:x 6040:( 6038:f 6028:x 6020:x 6018:( 6016:f 5994:x 5992:( 5986:x 5982:x 5978:x 5974:x 5970:x 5966:x 5962:x 5960:( 5958:f 5937:x 5933:x 5926:x 5922:x 5918:x 5914:x 5910:x 5908:( 5906:f 5885:x 5883:( 5879:x 5875:x 5871:x 5867:x 5863:x 5861:( 5859:f 5836:x 5832:x 5828:x 5824:x 5820:x 5818:( 5816:f 5795:x 5791:x 5787:x 5779:x 5775:x 5771:x 5767:x 5765:( 5763:f 5742:x 5738:x 5730:x 5726:x 5722:x 5720:( 5718:f 5697:x 5693:x 5691:( 5689:f 5668:x 5666:( 5664:f 5630:. 5625:0 5621:a 5617:+ 5614:x 5611:) 5606:1 5602:a 5598:+ 5595:x 5592:) 5587:2 5583:a 5579:+ 5576:x 5573:) 5568:3 5564:a 5560:+ 5554:+ 5551:x 5548:) 5543:2 5537:n 5533:a 5529:+ 5526:x 5523:) 5518:1 5512:n 5508:a 5504:+ 5501:x 5496:n 5492:a 5488:( 5485:( 5482:( 5479:( 5476:( 5424:] 5421:1 5418:, 5415:1 5409:[ 5387:2 5383:x 5376:1 5356:, 5351:2 5346:) 5339:2 5335:x 5328:1 5323:( 5293:2 5289:y 5285:+ 5280:5 5276:y 5272:x 5269:+ 5266:y 5261:2 5257:x 5253:4 5250:+ 5245:3 5241:x 5237:2 5234:= 5231:) 5228:y 5225:, 5222:x 5219:( 5216:f 5196:, 5193:x 5185:3 5181:x 5177:= 5174:) 5171:x 5168:( 5165:f 5154:f 5131:n 5129:a 5125:2 5122:a 5118:1 5115:a 5111:0 5108:a 5102:n 5097:f 5089:x 5075:) 5072:x 5069:( 5066:f 5044:0 5040:a 5036:+ 5033:x 5028:1 5024:a 5020:+ 5015:2 5011:x 5005:2 5001:a 4997:+ 4991:+ 4986:1 4980:n 4976:x 4970:1 4964:n 4960:a 4956:+ 4951:n 4947:x 4941:n 4937:a 4921:f 4894:1 4889:x 4885:x 4879:p 4872:k 4868:a 4863:k 4857:k 4847:p 4831:c 4827:x 4823:x 4817:3 4814:/ 4811:1 4801:x 4796:c 4779:1 4776:+ 4773:i 4766:1 4763:+ 4760:i 4756:x 4750:i 4746:a 4737:n 4732:0 4729:= 4726:i 4718:+ 4715:c 4712:= 4709:c 4706:+ 4703:x 4698:0 4694:a 4690:+ 4685:2 4679:2 4675:x 4669:1 4665:a 4658:+ 4653:3 4647:3 4643:x 4637:2 4633:a 4626:+ 4620:+ 4615:n 4609:n 4605:x 4599:1 4593:n 4589:a 4582:+ 4576:1 4573:+ 4570:n 4563:1 4560:+ 4557:n 4553:x 4547:n 4543:a 4519:P 4495:. 4490:1 4484:i 4480:x 4474:i 4470:a 4466:i 4461:n 4456:1 4453:= 4450:i 4442:= 4437:1 4433:a 4429:+ 4426:x 4421:2 4417:a 4413:2 4410:+ 4404:+ 4399:2 4393:n 4389:x 4383:1 4377:n 4373:a 4369:) 4366:1 4360:n 4357:( 4354:+ 4349:1 4343:n 4339:x 4333:n 4329:a 4325:n 4315:x 4299:i 4295:x 4289:i 4285:a 4279:n 4274:0 4271:= 4268:i 4260:= 4255:0 4251:a 4247:+ 4244:x 4239:1 4235:a 4231:+ 4226:2 4222:x 4216:2 4212:a 4208:+ 4202:+ 4197:1 4191:n 4187:x 4181:1 4175:n 4171:a 4167:+ 4162:n 4158:x 4152:n 4148:a 4144:= 4141:P 4080:) 4074:2 4068:3 4063:i 4057:1 4051:+ 4048:x 4044:( 4039:) 4033:2 4027:3 4022:i 4019:+ 4016:1 4010:+ 4007:x 4003:( 3999:) 3996:1 3990:x 3987:( 3984:5 3963:) 3959:1 3956:+ 3953:x 3950:+ 3945:2 3941:x 3936:( 3932:) 3929:1 3923:x 3920:( 3917:5 3897:5 3889:3 3885:x 3881:5 3835:) 3833:c 3831:( 3829:a 3821:) 3819:x 3817:( 3815:b 3810:) 3808:x 3806:( 3804:a 3795:c 3790:c 3786:x 3782:x 3780:( 3778:b 3769:) 3767:x 3765:( 3763:b 3747:) 3745:b 3741:r 3734:r 3730:q 3727:b 3723:a 3718:) 3716:x 3714:( 3712:r 3704:) 3702:x 3700:( 3698:q 3690:) 3688:x 3686:( 3684:b 3680:x 3678:( 3676:a 3660:x 3654:x 3606:. 3603:) 3600:2 3597:+ 3594:x 3591:3 3588:( 3585:2 3582:+ 3577:2 3573:) 3569:2 3566:+ 3563:x 3560:3 3557:( 3554:= 3551:) 3548:) 3545:x 3542:( 3539:g 3536:( 3533:f 3530:= 3527:) 3524:x 3521:( 3518:) 3515:g 3509:f 3506:( 3486:2 3483:+ 3480:x 3477:3 3474:= 3471:) 3468:x 3465:( 3462:g 3442:x 3439:2 3436:+ 3431:2 3427:x 3423:= 3420:) 3417:x 3414:( 3411:f 3391:g 3385:f 3372:g 3358:f 3327:+ 3324:y 3318:+ 3313:2 3309:y 3305:x 3302:3 3299:+ 3294:2 3290:y 3283:+ 3280:x 3274:+ 3271:y 3266:2 3262:x 3258:2 3255:+ 3252:y 3249:x 3243:+ 3238:2 3234:x 3230:4 3227:= 3224:Q 3221:P 3197:5 3192:+ 3187:) 3184:y 3178:+ 3175:y 3172:3 3169:( 3164:+ 3157:2 3153:y 3149:x 3146:3 3141:+ 3134:2 3130:y 3121:+ 3112:) 3109:x 3103:+ 3100:x 3097:2 3094:( 3089:+ 3084:y 3079:2 3075:x 3071:2 3066:+ 3061:) 3058:y 3055:x 3052:5 3049:+ 3046:y 3043:x 3040:6 3037:+ 3034:y 3031:x 3025:( 3020:+ 3013:2 3009:x 3005:4 2999:= 2994:Q 2991:P 2958:+ 2953:y 2950:x 2947:5 2942:+ 2937:y 2929:+ 2924:x 2916:+ 2907:y 2904:3 2899:+ 2892:2 2888:y 2884:x 2881:3 2876:+ 2869:2 2865:y 2856:+ 2851:y 2848:x 2845:6 2840:+ 2831:x 2828:2 2823:+ 2818:y 2813:2 2809:x 2805:2 2800:+ 2795:y 2792:x 2784:+ 2777:2 2773:x 2769:4 2763:= 2758:Q 2755:P 2727:) 2721:1 2709:5 2703:( 2698:+ 2693:) 2687:y 2684:x 2672:5 2666:( 2661:+ 2656:) 2650:y 2647:5 2635:5 2629:( 2624:+ 2619:) 2613:x 2610:2 2598:5 2592:( 2587:+ 2578:) 2572:1 2560:y 2557:3 2551:( 2546:+ 2541:) 2535:y 2532:x 2520:y 2517:3 2511:( 2506:+ 2501:) 2495:y 2492:5 2480:y 2477:3 2471:( 2466:+ 2461:) 2455:x 2452:2 2440:y 2437:3 2431:( 2426:+ 2417:) 2411:1 2399:x 2396:2 2390:( 2385:+ 2380:) 2374:y 2371:x 2359:x 2356:2 2350:( 2345:+ 2340:) 2334:y 2331:5 2319:x 2316:2 2310:( 2305:+ 2300:) 2294:x 2291:2 2279:x 2276:2 2270:( 2263:= 2254:Q 2245:P 2212:1 2209:+ 2206:y 2203:x 2200:+ 2197:y 2194:5 2191:+ 2188:x 2185:2 2182:= 2174:Q 2164:5 2161:+ 2158:y 2155:3 2152:+ 2149:x 2146:2 2143:= 2135:P 2086:+ 2081:2 2077:y 2073:4 2070:+ 2067:y 2064:x 2061:5 2058:+ 2055:x 2052:= 2049:Q 2046:+ 2043:P 2023:) 2020:2 2014:8 2011:( 2008:+ 2003:2 1999:y 1995:4 1992:+ 1989:y 1986:x 1983:5 1980:+ 1977:) 1974:x 1971:3 1968:+ 1965:x 1962:2 1956:( 1953:+ 1950:) 1945:2 1941:x 1937:3 1929:2 1925:x 1921:3 1918:( 1915:= 1912:Q 1909:+ 1906:P 1886:8 1883:+ 1878:2 1874:y 1870:4 1867:+ 1864:x 1861:3 1858:+ 1853:2 1849:x 1845:3 1839:2 1833:y 1830:x 1827:5 1824:+ 1821:x 1818:2 1810:2 1806:x 1802:3 1799:= 1796:Q 1793:+ 1790:P 1770:8 1767:+ 1762:2 1758:y 1754:4 1751:+ 1748:x 1745:3 1742:+ 1737:2 1733:x 1729:3 1723:= 1720:Q 1700:2 1694:y 1691:x 1688:5 1685:+ 1682:x 1679:2 1671:2 1667:x 1663:3 1660:= 1657:P 1625:z 1619:y 1615:x 1518:2 1513:x 1508:3 1503:x 1496:x 1492:x 1490:3 1485:x 1479:x 1462:x 1458:y 1455:x 1451:y 1448:x 1441:n 1428:n 1414:x 1408:x 1406:( 1404:f 1386:x 1382:x 1376:x 1374:2 1328:. 1318:3 1309:m 1306:r 1303:e 1300:t 1285:4 1278:+ 1263:2 1254:m 1251:r 1248:e 1245:t 1230:x 1227:5 1205:1 1196:m 1193:r 1190:e 1187:t 1170:2 1166:x 1162:3 1133:y 1127:x 1121:y 1115:x 1096:y 1091:2 1087:x 1083:5 1055:x 1051:x 1011:k 1007:x 1001:k 997:a 991:n 986:0 983:= 980:k 940:x 920:x 894:n 890:a 886:, 880:, 875:0 871:a 850:, 845:0 841:a 837:+ 834:x 829:1 825:a 821:+ 816:2 812:x 806:2 802:a 798:+ 792:+ 787:1 781:n 777:x 771:1 765:n 761:a 757:+ 752:n 748:x 742:n 738:a 726:x 708:2 705:+ 702:x 699:3 691:2 687:x 683:= 680:) 677:2 671:x 668:( 665:) 662:1 656:x 653:( 630:2 627:+ 624:x 621:3 613:2 609:x 588:) 585:2 579:x 576:( 573:) 570:1 564:x 561:( 473:, 470:P 467:= 464:) 461:x 458:( 455:P 445:x 441:x 437:P 433:x 425:x 421:a 414:a 412:( 410:P 406:a 398:a 394:P 376:, 373:) 370:a 367:( 364:P 358:a 348:P 344:P 340:x 336:a 332:a 330:( 328:P 324:a 317:x 313:P 309:x 307:( 305:P 301:x 299:( 297:P 290:x 288:( 286:P 282:P 278:x 276:( 274:P 270:P 266:x 262:P 255:x 247:x 235:x 101:x 94:x 90:x 84:x 34:. 20:)

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Index