In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x is x² − 4x + 7. An example with three indeterminates is x³ + 2xyz² − yz + 1. Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms. The word polynomial was first used in the 17th century. The x occurring in a polynomial is commonly called a variable or an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value. However, when one considers the function defined by the polynomial, then x represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably. A polynomial P in the indeterminate x is commonly denoted either as P or as P. Formally, the name of the polynomial is P, not P, but the use of the functional notation P 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 P be a polynomial" is a shorthand for "let P be a polynomial in the indeterminate x". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name of the indeterminate do not appear at each occurrence of the polynomial. The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials. If a denotes a number, a variable, another polynomial, or, more generally, any expression, then P(a) denotes, by convention, the result of substituting a for x in P. Thus, the polynomial P defines the function displaystyleamapstoP(a), which is the polynomial function associated to P. Frequently, when using this notation, one supposes that a is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any ring). In particular, if a is a polynomial then P(a) is also a polynomial. More specifically, when a is the indeterminate x, then the image of x by this function is the polynomial P itself (substituting x for x does not change anything). In other words, displaystyleP(x)=P, which justifies formally the existence of two notations for the same polynomial. A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any expression that do not involve the indeterminates, and represent mathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining the same polynomial if they may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication. For example displaystyle(x-1)(x-2) and displaystylex²-3x+2 are two polynomial expressions that represent the same polynomial; so, one has the equality displaystyle(x-1)(x-2)=x²-3x+2. A polynomial in a single indeterminate x can always be written (or rewritten) in the form displaystyleaₙxⁿ+aₙ₋₁xⁿ⁻¹+dotsb+a₂x²+a₁x+a₀, where displaystylea₀,ldots,aₙ are constants that are called the coefficients of the polynomial, and displaystyle x is the indeterminate. The word "indeterminate" means that displaystyle x represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function. This can be expressed more concisely by using summation notation: displaystylesumₖ₌₀ⁿaₖxᵏ That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term – and a finite number of indeterminates, raised to non-negative integer powers. The exponent on an indeterminate in a term is called the degree of that indeterminate in that term; the degree of the term is the sum of the degrees of the indeterminates in that term, and the degree of a polynomial is the largest degree of any term with nonzero coefficient. Because x = x¹, the degree of an indeterminate without a written exponent is one. A term with no indeterminates and a polynomial with no indeterminates are called, respectively, a constant term and a constant polynomial. The degree of a constant term and of a nonzero constant polynomial is 0. The degree of the zero polynomial 0 is generally treated as not defined. For example: displaystyle-5x²y is a term. The coefficient is −5, the indeterminates are x and y, the degree of x is two, while the degree of y is one. The degree of the entire term is the sum of the degrees of each indeterminate in it, so in this example the degree is 2 + 1 = 3. Forming a sum of several terms produces a polynomial. For example, the following is a polynomial: displaystyleunderbrace,3x²bₑgᵢₙₛₘₐₗₗₘₐₜᵣᵢₓₘₐₜₕᵣₘₜₑᵣₘ₁ₑₙdₛₘₐₗₗₘₐₜᵣᵢₓunderbrace-,5xbₑgᵢₙₛₘₐₗₗₘₐₜᵣᵢₓₘₐₜₕᵣₘₜₑᵣₘ₂ₑₙdₛₘₐₗₗₘₐₜᵣᵢₓunderbrace+,4bₑgᵢₙₛₘₐₗₗₘₐₜᵣᵢₓₘₐₜₕᵣₘₜₑᵣₘ₃ₑₙdₛₘₐₗₗₘₐₜᵣᵢₓ. It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Polynomials of small degree have been given specific names. A polynomial of degree zero is a constant polynomial, or simply a constant. Polynomials of degree one, two or three are respectively linear polynomials, quadratic polynomials and cubic polynomials. For higher degrees, the specific names are not commonly used, although quartic polynomial and quintic polynomial are sometimes used. The names for the degrees may be applied to the polynomial or to its terms. For example, the term 2x in x² + 2x + 1 is a linear term in a quadratic polynomial. The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather, the degree of the zero polynomial is either left explicitly undefined, or defined as negative. The zero polynomial is also unique in that it is the only polynomial in one indeterminate that has an infinite number of roots. The graph of the zero polynomial, f = 0, is the x-axis. In the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. For example, x³y² + 7x²y³ − 3x⁵ is homogeneous of degree 5. For more details, see Homogeneous polynomial. The commutative law of addition can be used to rearrange terms into any preferred order. In polynomials with one indeterminate, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial 3x² − 5x + 4 is written in descending powers of x. The first term has coefficient 3, indeterminate x, and exponent 2. In the second term, the coefficient is −5. The third term is a constant. Because the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two. Two terms with the same indeterminates raised to the same powers are called "similar terms" or "like terms", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0. Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. A real polynomial is a polynomial with real coefficients. When it is used to define a function, the domain is not so restricted. However, a real polynomial function is a function from the reals to the reals that is defined by a real polynomial. Similarly, an integer polynomial is a polynomial with integer coefficients, and a complex polynomial is a polynomial with complex coefficients. A polynomial in one indeterminate is called a univariate polynomial, a polynomial in more than one indeterminate is called a multivariate polynomial. A polynomial with two indeterminates is called a bivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance, when working with univariate polynomials, one does not exclude constant polynomials, although strictly speaking, constant polynomials do not contain any indeterminates at all. It is possible to further classify multivariate polynomials as bivariate, trivariate, and so on, according to the maximum number of indeterminates allowed. Again, so that the set of objects under consideration be closed under subtraction, a study of trivariate polynomials usually allows bivariate polynomials, and so on. It is also common to say simply "polynomials in x, y, and z", listing the indeterminates allowed. Polynomials can be added using the associative law of addition (grouping all their terms together into a single sum), possibly followed by reordering (using the commutative law) and combining of like terms. For example, if displaystyleP=3x²-2x+5xy-2 and displaystyleQ=-3x²+3x+4y²+8 then the sum displaystyleP+Q=3x²-2x+5xy-2-3x²+3x+4y²+8 can be reordered and regrouped as displaystyleP+Q=(3x²-3x²)+(-2x+3x)+5xy+4y²+(8-2) and then simplified to displaystyleP+Q=x+5xy+4y²+6. When polynomials are added together, the result is another polynomial. Subtraction of polynomials is similar. Polynomials can also be multiplied. To expand the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if displaystylebeginalignedcolorRedP&colorRed=2x+3y+5BlueQ&colorBlue=2x+5y+xy+1endaligned then displaystylebeginarrayrccrcrcrcrcolorRedPcolorBlueQ&=&&(colorRed2xcdotcolorBlue2x)&+&(colorRed2xcdotcolorBlue5y)&+&(colorRed2xcdotcolorBluexy)&+&(colorRed2xcdotcolorBlue1)&&+&(colorRed3ycdotcolorBlue2x)&+&(colorRed3ycdotcolorBlue5y)&+&(colorRed3ycdotcolorBluexy)&+&(colorRed3ycdotcolorBlue1)&&+&(colorRed5cdotcolorBlue2x)&+&(colorRed5cdotcolorBlue5y)&+&(colorRed5cdotcolorBluexy)&+&(colorRed5cdotcolorBlue1)endarray Carrying out the multiplication in each term produces displaystylebeginarrayrccrcrcrcrPQ&=&&4x²&+&10xy&+&2x²y&+&2x&&+&6xy&+&15y²&+&3xy²&+&3y&&+&10x&+&25y&+&5xy&+&5.endarray Combining similar terms yields displaystylebeginarrayrcccrcrcrcrPQ&=&&4x²&+&(10xy+6xy+5xy)&+&2x²y&+&(2x+10x)&&+&15y²&+&3xy²&+&(3y+25y)&+&5endarray which can be simplified to displaystylePQ=4x²+21xy+2x²y+12x+15y²+3xy²+28y+5. As in the example, the product of polynomials is always a polynomial. Given a polynomial displaystyle f of a single variable and another polynomial g of any number of variables, the composition displaystyle fcirc g is obtained by substituting each copy of the variable of the first polynomial by the second polynomial. For example, if displaystylef(x)=x²+2x and displaystyleg(x)=3x+2 then displaystyle(fcirc g)(x)=f(g(x))=(3x+2)²+2(3x+2). 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. The division of one polynomial by another is not typically a polynomial. Instead, such ratios are a more general family of objects, called rational fractions, rational expressions, or rational functions, depending on context. This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer. For example, the fraction 1/ is not a polynomial, and it cannot be written as a finite sum of powers of the variable x. For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers. This notion of the division a/b results in two polynomials, a quotient q and a remainder r, such that a = b q + r and degree < degree. The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division. When the denominator b is monic and linear, that is, b = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a by b is the evaluation a. In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division. All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear. Over the real numbers, they have the degree either one or two. Over the integers and the rational numbers the irreducible factors may have any degree. For example, the factored form of displaystyle5x³-5 is displaystyle5(x-1)left(x²+x+1right) over the integers and the reals, and displaystyle5(x-1)left(x+frac1+isqrt32right)left(x+frac1-isqrt32right) over the complex numbers. The computation of the factored form, called factorization is, in general, too difficult to be done by hand-written computation. However, efficient polynomial factorization algorithms are available in most computer algebra systems. Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions. The derivative of the polynomial displaystyleP=aₙxⁿ+aₙ₋₁xⁿ⁻¹+dots+a₂x²+a₁x+a₀=sumᵢ₌₀ⁿaᵢxⁱ with respect to x is the polynomial displaystylenaₙxⁿ⁻¹+(n-1)aₙ₋₁xⁿ⁻²+dots+2a₂x+a₁=sumᵢ₌₁ⁿiaᵢxⁱ⁻¹. Similarly, the general antiderivative (or indefinite integral) of displaystyle P is displaystylefracaₙxⁿ⁺¹n+1+fracaₙ₋₁xⁿn+dots+fraca₂x³3+fraca₁x²2+a₀x+c=c+sumᵢ₌₀ⁿfracaᵢxⁱ⁺¹i+1 where c is an arbitrary constant. For example, antiderivatives of x² + 1 have the form ⁠1/3⁠x³ + x + c. For polynomials whose coefficients come from more abstract settings, the formula for the derivative can still be interpreted formally, with the coefficient kaₖ understood to mean the sum of k copies of aₖ. For example, over the integers modulo p, the derivative of the polynomial x + x is the polynomial 1. A polynomial function is a function that can be defined by evaluating a polynomial. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial displaystyleaₙxⁿ+aₙ₋₁xⁿ⁻¹+cdots+a₂x²+a₁x+a₀ that evaluates to displaystylef(x) for all x in the domain of f (here, n is a non-negative integer and a₀, a₁, a₂, ..., aₙ are constant coefficients). Generally, unless otherwise specified, polynomial functions have complex coefficients, arguments, and values. In particular, a polynomial, restricted to have real coefficients, defines a function from the complex numbers to the complex numbers. If the domain of this function is also restricted to the reals, the resulting function is a real function that maps reals to reals. For example, the function f, defined by displaystylef(x)=x³-x, is a polynomial function of one variable. Polynomial functions of several variables are similarly defined, using polynomials in more than one indeterminate, as in displaystylef(x,y)=2x³+4x²y+xy⁵+y²-7. 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 displaystyleleft(sqrt1-x²right)², which takes the same values as the polynomial displaystyle1-x² on the interval displaystyle[-1,1], and thus both expressions define the same polynomial function on this interval. Every polynomial function is continuous, smooth, and entire. The evaluation 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. For polynomials in one indeterminate, the evaluation is usually more efficient (lower number of arithmetic operations to perform) using Horner's method, which consists of rewriting the polynomial as displaystyle(((((aₙx+aₙ₋₁)x+aₙ₋₂)x+dotsb+a₃)x+a₂)x+a₁)x+a₀. A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f = 0 is the x-axis. The graph of a degree 0 polynomial f = a₀, where a₀ ≠ 0, is a horizontal line with y-intercept a₀ The graph of a degree 1 polynomial f = a₀ + a₁x, where a₁ ≠ 0, is an oblique line with y-intercept a₀ and slope a₁. The graph of a degree 2 polynomial f = a₀ + a₁x + a₂x², where a₂ ≠ 0 is a parabola. The graph of a degree 3 polynomial f = a₀ + a₁x + a₂x² + a₃x³, where a₃ ≠ 0 is a cubic curve. The graph of any polynomial with degree 2 or greater f = a₀ + a₁x + a₂x² + ⋯ + aₙx, where aₙ ≠ 0 and n ≥ 2 is a continuous non-linear curve. A non-constant polynomial function tends to infinity when the variable increases indefinitely. If the degree is higher than one, the graph does not have any asymptote. It has two parabolic branches with vertical direction. Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. A polynomial equation, also called an algebraic equation, is an equation of the form displaystyleaₙxⁿ+aₙ₋₁xⁿ⁻¹+dotsb+a₂x²+a₁x+a₀=0. For example, displaystyle3x²+4x-5=0 is a polynomial equation. When considering equations, the indeterminates of polynomials are also called unknowns, and the solutions are the possible values of the unknowns for which the equality is true. A polynomial equation stands in contrast to a polynomial identity like = x² − y², where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality. In elementary algebra, methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the cubic and quartic equations. For higher degrees, the Abel–Ruffini theorem asserts that there can not exist a general formula in radicals. However, root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree. The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the complex solutions are counted with their multiplicity. This fact is called the fundamental theorem of algebra. A root of a nonzero univariate polynomial P is a value a of x such that P(a) = 0. In other words, a root of P is a solution of the polynomial equation P = 0 or a zero of the polynomial function defined by P. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered. A number a is a root of a polynomial P if and only if the linear polynomial x − a divides P, that is if there is another polynomial Q such that P = Q. It may happen that a power of x − a divides P; in this case, a is a multiple root of P, and otherwise a is a simple root of P. If P is a nonzero polynomial, there is a highest power m such that divides P, which is called the multiplicity of a as a root of P. The number of roots of a nonzero polynomial P, counted with their respective multiplicities, cannot exceed the degree of P, and equals this degree if all complex roots are considered. The coefficients of a polynomial and its roots are related by Vieta's formulas. Some polynomials, such as x² + 1, do not have any roots among the real numbers. If, however, the set of accepted solutions is expanded to the complex numbers, every non-constant polynomial has at least one root; this is the fundamental theorem of algebra. By successively dividing out factors x − a, one sees that any polynomial with complex coefficients can be written as a constant times a product of such polynomial factors of degree 1; as a consequence, the number of roots counted with their multiplicities is exactly equal to the degree of the polynomial. There may be several meanings of "solving an equation". One may want to express the solutions as explicit numbers; for example, the unique solution of 2x − 1 = 0 is 1/2. This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expressions; for example, the golden ratio displaystyle(1+sqrt5)/2 is the unique positive solution of displaystylex²-x-1=0. In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see cubic equation and quartic equation). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, Niels Henrik Abel proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see Abel–Ruffini theorem). In 1830, Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of Galois theory and group theory, two important branches of modern algebra. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation). When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to compute numerical approximations of the solutions. There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily polynomial equations of degree higher than 1,000. 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 zeros instead of "roots". The study of the sets of zeros of polynomials is the object of algebraic geometry. For a set of polynomial equations with several unknowns, there are algorithms to decide whether they have a finite number of complex solutions, and, if this number is finite, for computing the solutions. See System of polynomial equations. The special case where all the polynomials are of degree one is called a system of linear equations, for which another range of different solution methods exist, including the classical Gaussian elimination. A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general algorithm for solving them, or even for deciding whether the set of solutions is empty. Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as Fermat's Last Theorem. Polynomials where indeterminates are substituted for some other mathematical objects are often considered, and sometimes have a special name. A trigonometric polynomial is a finite linear combination of functions sin and cos with n taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. If sin and cos are expanded in terms of sin and cos, a trigonometric polynomial becomes a polynomial in the two variables sin and cos. Conversely, every polynomial in sin and cos may be converted, with Product-to-sum identities, into a linear combination of functions sin and cos. This equivalence explains why linear combinations are called polynomials. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are also used in the discrete Fourier transform. A matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial displaystyleP(x)=sumᵢ₌₀ⁿaᵢxⁱ=a₀+a₁x+a₂x²+cdots+aₙxⁿ, this polynomial evaluated at a matrix A is displaystyleP(A)=sumᵢ₌₀ⁿaᵢAⁱ=a₀I+a₁A+a₂A²+cdots+aₙAⁿ, where I is the identity matrix. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring Mₙ. A bivariate polynomial where the second variable is substituted for an exponential function applied to the first variable, for example P, may be called an exponential polynomial. A rational fraction is the quotient of two polynomials. Any algebraic expression that can be rewritten as a rational fraction is a rational function. While polynomial functions are defined for all values of the variables, a rational function is defined only for the values of the variables for which the denominator is not zero. The rational fractions include the Laurent polynomials, but do not limit denominators to powers of an indeterminate. Laurent polynomials are like polynomials, but allow negative powers of the variable to occur. Formal power series are like polynomials, but allow infinitely many non-zero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down, but the rules for manipulating their terms are the same as for polynomials. Non-formal power series also generalize polynomials, but the multiplication of two power series may not converge. A polynomial f over a commutative ring R is a polynomial all of whose coefficients belong to R. It is straightforward to verify that the polynomials in a given set of indeterminates over R form a commutative ring, called the polynomial ring in these indeterminates, denoted displaystyleR[x] in the univariate case and displaystyleR[x₁,ldots,xₙ] in the multivariate case. One has displaystyleR[x₁,ldots,xₙ]=left(R[x₁,ldots,xₙ₋₁]right)[xₙ]. So, most of the theory of the multivariate case can be reduced to an iterated univariate case. The map from R to R sending r to itself considered as a constant polynomial is an injective ring homomorphism, by which R is viewed as a subring of R. In particular, R is an algebra over R. One can think of the ring R as arising from R by adding one new element x to R, and extending in a minimal way to a ring in which x satisfies no other relations than the obligatory ones, plus commutation with all elements of R. To do this, one must add all powers of x and their linear combinations as well. Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial x² + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R. If R is commutative, then one can associate with every polynomial P in R a polynomial function f with domain and range equal to R. One obtains the value f by substitution of the value r for the symbol x in P. One reason to distinguish between polynomials and polynomial functions is that, over some rings, different polynomials may give rise to the same polynomial function. This is not the case when R is the real or complex numbers, whence the two concepts are not always distinguished in analysis. An even more important reason to distinguish between polynomials and polynomial functions is that many operations on polynomials require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for x. If R is an integral domain and f and g are polynomials in R[x], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[x] such that f q = g. If displaystyleainR, then a is a root of f if and only displaystylex-a divides f. In this case, the quotient can be computed using the polynomial long division. If F is a field and f and g are polynomials in F[x] with g ≠ 0, then there exist unique polynomials q and r in F[x] with displaystylef=q,g+r and such that the degree of r is smaller than the degree of g (using the convention that the polynomial 0 has a negative degree). The polynomials q and r are uniquely determined by f and g. This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant polynomials. In the case of coefficients in a ring, "non-constant" must be replaced by "non-constant or non-unit". Any polynomial may be decomposed into the product of an invertible constant by a product of irreducible polynomials. If the coefficients belong to a field or a unique factorization domain this decomposition is unique up to the order of the factors and the multiplication of any non-unit factor by a unit. When the coefficients belong to integers, rational numbers or a finite field, there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials. These algorithms are not practicable for hand-written computation, but are available in any computer algebra system. Eisenstein's criterion can also be used in some cases to determine irreducibility. In modern positional numbers systems, such as the decimal system, the digits and their positions in the representation of an integer, for example, 45, are a shorthand notation for a polynomial in the radix or base, in this case, 4 × 10¹ + 5 × 10⁰. As another example, in radix 5, a string of digits such as 132 denotes the number 1 × 5² + 3 × 5¹ + 2 × 5⁰ = 42. This representation is unique. Let b be a positive integer greater than 1. Then every positive integer a can be expressed uniquely in the form displaystylea=rₘbᵐ+rₘ₋₁bᵐ⁻¹+dotsb+r₁b+r₀, where m is a nonnegative integer and the r's are integers such that 0 < rₘ < b and 0 ≤ rᵢ < b for i = 0, 1,..., m − 1. The simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. An important example in calculus is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, and the Stone–Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial function. Practical methods of approximation include polynomial interpolation and the use of splines. Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph. The term "polynomial", as an adjective, can also be used for quantities or functions that can be written in polynomial form. For example, in computational complexity theory the phrase polynomial time means that the time it takes to complete an algorithm is bounded by a polynomial function of some variable, such as the size of the input. Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations were written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, c. 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29. The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the as denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well. List of polynomial topics Markushevich, A.I. , "Polynomial", Encyclopedia of Mathematics, EMS Press "Euler's Investigations on the Roots of Equations". Archived from the original on September 24, 2012.

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as displaystyleax²+bx+c=0,, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term. The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation displaystyleax²+bx+c=a(x-r)(x-s)=0 where r and s are the solutions for x. The quadratic formula displaystylex=frac-bpmsqrtb²-4ac2a expresses the solutions in terms of a, b, and c. Completing the square is one of several ways for deriving the formula. Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two. A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots. When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root. When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part, and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct. The solutions of a quadratic equation can be found by several alternative methods. It may be possible to express a quadratic equation ax² + bx + c = 0 as a product = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic. For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form x² + bx + c = 0, the sought factorization has the form, and one has to find two numbers q and s that add up to b and whose product is c. As an example, x² + 5x + 6 factors as. The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the algebraic identity displaystylex²+2hx+h²=(x+h)², which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, ax² + bx + c = 0 Divide each side by a, the coefficient of the squared term. Subtract the constant term c/a from both sides. Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square. Write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side. Solve each of the two linear equations. We illustrate use of this algorithm by solving 2x² + 4x − 4 = 0 displaystyle2x²+4x-4=0 displaystylex²+2x-2=0 displaystylex²+2x=2 displaystylex²+2x+1=2+1 displaystyleleft(x+1right)²=3 displaystylex+1=pmsqrt3 displaystylex=-1pmsqrt3 The plus–minus symbol "±" indicates that both textstylex=-1+sqrt3 and textstylex=-1-sqrt3 are solutions of the quadratic equation. Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula. The mathematical proof will now be briefly summarized. It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: displaystyleleft(x+fracb2aright)²=fracb²-4ac4a². Taking the square root of both sides, and isolating x, gives: displaystylex=frac-bpmsqrtb²-4ac2a. Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax² + 2bx + c = 0 or ax² − 2bx + c = 0, where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent. A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics. A lesser known quadratic formula, as used in Muller's method, provides the same roots via the equation displaystylex=frac2c-bpmsqrtb²-4ac. This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a. It also follows from dividing the quadratic equation by displaystylex² giving displaystylecx⁻²+bx⁻¹+a=0, solving this for displaystylex⁻¹, and then inverting. One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0. When neither a nor c is zero, the equality between the standard quadratic formula and Muller's method, displaystylefrac2c-b-sqrtb²-4ac=frac-b+sqrtb²-4ac2a,, can be verified by cross multiplication, and similarly for the other choice of signs. It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation: displaystylex²+px+q=0, where p = b/a and q = c/a. This monic polynomial equation has the same solutions as the original. The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is displaystylex=-fracp2pmsqrtleft(fracp2right)²-q,. In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta: displaystyleDelta=b²-4ac. A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases: If the discriminant is positive, then there are two distinct roots displaystylefrac-b+sqrtDelta2aquadtextandquadfrac-b-sqrtDelta2a, both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one real root displaystyle-fracb2a, sometimes called a repeated or double root or two equal roots. If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex rootsdisplaystyle-fracb2a+ifracsqrt-Delta2aquadtextandquad-fracb2a-ifracsqrt-Delta2a, which are complex conjugates of each other. In these expressions i is the imaginary unit. Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. The function f(x) = ax² + bx + c is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. If a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at displaystylescriptstylex=tfrac-b2a, and the y-coordinate of the vertex may be found by substituting this x-value into the function. The y-intercept is located at the point (0, c). The solutions of the quadratic equation ax² + bx + c = 0 correspond to the roots of the function f = ax² + bx + c, since they are the values of x for which f = 0. If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. If the discriminant is positive, the graph touches the x-axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x-axis. The term displaystylex-r is a factor of the polynomial displaystyleax²+bx+c if and only if r is a root of the quadratic equation displaystyleax²+bx+c=0. It follows from the quadratic formula that displaystyleax²+bx+c=aleft(x-frac-b+sqrtb²-4ac2aright)left(x-frac-b-sqrtb²-4ac2aright). In the special case b² = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as displaystyleax²+bx+c=aleft(x+fracb2aright)². The solutions of the quadratic equation displaystyleax²+bx+c=0 may be deduced from the graph of the quadratic function displaystylef(x)=ax²+bx+c, which is a parabola. If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points. If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola. If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be. Let h and k be respectively the x-coordinate and the y-coordinate of the vertex of the parabola (that is the point with maximal or minimal y-coordinate. The quadratic function may be rewritten displaystyley=a(x-h)²+k. Let d be the distance between the point of y-coordinate 2k on the axis of the parabola, and a point on the parabola with the same y-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is h, and their imaginary part are ±d. That is, the roots are displaystyleh+idquadtextandquadh-id, or in the case of the example of the figure displaystyle5+3iquadtextandquad5-3i. Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers. In this context, the quadratic formula is not completely stable. This occurs when the roots have different order of magnitude, or, equivalently, when b² and b² − 4ac are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, r, can be computed as displaystyle(c/a)/R where R is the root that is bigger in magnitude. This is equivalent to using the formula displaystylex=frac-2cbpmsqrtb²-4ac using the plus sign if displaystyleb>0 and the minus sign if displaystyleb<0. A second form of cancellation can occur between the terms b² and 4ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots. The golden ratio is found as the positive solution of the quadratic equation displaystylex²-x-1=0. The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables. Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. Descartes' theorem states that for every four kissing circles, their radii satisfy a particular quadratic equation. The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral. Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation. In physics, for motion with constant acceleration displaystyle a, the displacement or position displaystyle x of a moving body can be expressed as a quadratic function of time displaystyle t given the initial position displaystylex₀ and initial velocity displaystylev₀: textstylex=x₀+v₀t+frac12at². In chemistry, the pH of a solution of weak acid can be calculated from the negative base-10 logarithm of the positive root of a quadratic equation in terms of the acidity constant and the analytical concentration of the acid. Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: displaystylex+y=p,xy=q, which is equivalent to the statement that x and y are the roots of the equation: displaystylez²+q=pz. The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y, were as follows: Compute half of p. Square the result. Subtract q. Find the square root using a table of squares. Add together the results of steps and to give x. In modern notation this means calculating displaystylex=fracp2+sqrtleft(fracp2right)²-q, which is equivalent to the modern day quadratic formula for the larger real root (if any) displaystylex=frac-b+sqrtb²-4ac2a with a = 1, b = −p, and c = q. Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom, contains the solution to a two-term quadratic equation. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics. These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive. In 628 AD, Brahmagupta, an Indian mathematician, gave in his book Brāhmasphuṭasiddhānta the first explicit (although still not completely general) solution of the quadratic equation ax² + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." This is equivalent to displaystylex=fracsqrt4ac+b²-b2a. The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear indeterminate equations (originally of type ax/c = y). Muhammad ibn Musa al-Khwarizmi (9th century) developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. He also described the method of completing the square and recognized that the discriminant must be positive, which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions, as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations. The Jewish mathematician Abraham bar Hiyya Ha-Nasi authored the first European book to include the full solution to the general quadratic equation. His solution was largely based on Al-Khwarizmi's work. The writing of the Chinese mathematician Yang Hui is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. Vieta's formulas (named after François Viète) are the relations displaystylex₁+x₂=-fracba,quadx₁x₂=fracca between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation displaystyleleft(x-x₁right)left(x-x₂right)=x²-left(x₁+x₂right)x+x₁x₂=0 with the equation displaystylex²+fracbax+fracca=0. The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is displaystylexV=fracx₁+x₂2=-fracb2a. The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving displaystyleyV=-fracb²4a+c=-fracb²-4ac4a. Also, these formulas for the vertex can be deduced directly from the formula (see Completing the square) displaystyleax²+bx+c=aleft(x+fracb2aright)²-fracb²-4ac4a. For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If |x₂| << |x₁|, then x₁ + x₂ ≈ x₁, and we have the estimate: displaystylex₁approx-fracba. The second Vieta's formula then provides: displaystylex₂=fraccax₁approx-fraccb. These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. The figure shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse. This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation. In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots. Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations. It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation, where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting and then multiplying through by cos² / c, we obtain Introducing functions of 2θ and rearranging, we obtain where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation. Substituting the two values of θₙ or θₚ found from equations or into gives the required roots of. Complex roots occur in the solution based on equation if the absolute value of sin 2θₚ exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone. Calculating complex roots would require using a different trigonometric form. To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy: displaystyle4.16130x²+9.15933x-11.4207=0 A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries. displaystyleloga=0.6192290,logb=0.9618637,logc=1.0576927 displaystyle2sqrtac/b=2times10⁽⁰·⁶¹⁹²²⁹⁰⁺¹·⁰⁵⁷⁶⁹²⁷⁾/²⁻⁰·⁹⁶¹⁸⁶³⁷=1.505314 displaystyletheta=(tan⁻¹1.505314)/2=28.20169ᶜⁱʳᶜtextor-61.79831ᶜⁱʳᶜ displaystylelog|tantheta|=-0.2706462textor0.2706462 displaystylelogtextstylesqrtc/a=(1.0576927-0.6192290)/2=0.2192318 displaystylex₁=10⁰·²¹⁹²³¹⁸⁻⁰·²⁷⁰⁶⁴⁶²=0.888353 (rounded to six significant figures) displaystylex₂=-10⁰·²¹⁹²³¹⁸⁺⁰·²⁷⁰⁶⁴⁶²=-3.08943 If the quadratic equation displaystyleax²+bx+c=0 with real coefficients has two complex roots—the case where displaystyleb²-4ac<0, requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as displaystylex₁,,x₂=r(costhetapmisintheta), where displaystyler=sqrttfracca and displaystyletheta=cos⁻¹left(tfrac-b2sqrtacright). The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA. The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. The symbol displaystylepmsqrtb²-4ac in the formula should be understood as "either of the two elements whose square is b² − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field. In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial displaystylex²+bx+c over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is displaystylex=sqrtc and there is only one root since displaystyle-sqrtc=-sqrtc+2sqrtc=sqrtc. In summary, displaystyledisplaystylex²+c=(x+sqrtc)². See quadratic residue for more information about extracting square roots in finite fields. In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x² + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax² + bx + c are displaystylefracbaRleft(fracacb²right) and displaystylefracbaleft(Rleft(fracacb²right)+1right). For example, let a denote a multiplicative generator of the group of units of F₄, the Galois field of order four (thus a and a + 1 are roots of x² + x + 1 over F₄. Because (a + 1)² = a, a + 1 is the unique solution of the quadratic equation x² + a = 0. On the other hand, the polynomial x² + ax + 1 is irreducible over F₄, but it splits over F₁₆, where it has the two roots ab and ab + a, where b is a root of x² + x + a in F₁₆. This is a special case of Artin–Schreier theory. Solving quadratic equations with continued fractions Linear equation Cubic function Quartic equation Quintic equation Fundamental theorem of algebra "Quadratic equation", Encyclopedia of Mathematics, EMS Press, 2001 Weisstein, Eric W. "Quadratic equations". MathWorld.