Polynomial

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.[8][9] For example, if

${\displaystyle P=3x^{2}-2x+5xy-2}$ and ${\displaystyle Q=-3x^{2}+3x+4y^{2}+8}$

then the sum

${\displaystyle P+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8}$

can be reordered and regrouped as

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

and then simplified to

${\displaystyle P+Q=x+5xy+4y^{2}+6.}$

When polynomials are added together, the result is another polynomial.[10]

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.[8] For example, if

${\displaystyle {\begin{aligned}\color {Red}P&\color {Red}{=2x+3y+5}\\\color {Blue}Q&\color {Blue}{=2x+5y+xy+1}\end{aligned}}}$

then

${\displaystyle {\begin{array}{rccrcrcrcr}{\color {Red}{P}}{\color {Blue}{Q}}&{=}&&({\color {Red}{2x}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{2x}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{3y}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{3y}}\cdot {\color {Blue}{1}})\\&&+&({\color {Red}{5}}\cdot {\color {Blue}{2x}})&+&({\color {Red}{5}}\cdot {\color {Blue}{5y}})&+&({\color {Red}{5}}\cdot {\color {Blue}{xy}})&+&({\color {Red}{5}}\cdot {\color {Blue}{1}})\end{array}}}$

Carrying out the multiplication in each term produces

${\displaystyle {\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\end{array}}}$

Combining similar terms yields

${\displaystyle {\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}}$

which can be simplified to

${\displaystyle PQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5.}$

As in the example, the product of polynomials is always a polynomial.[10][5]

Given a polynomial ${\displaystyle f(x)}$ of a single variable and another polynomial g of any number of variables, the composition ${\displaystyle f\circ g}$ is obtained by substituting each copy of the variable of the first polynomial by the second polynomial.[5] For example, if ${\displaystyle f(x)=x^{2}+2x}$ and ${\displaystyle g(x)=3x+2}$ then

$${\displaystyle f\circ g=f(g(x))=(3x+2)^{2}+2(3x+2).}$$

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 always another polynomial.[11]

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.[12] This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer.[13][14] For example, the fraction 1/(x² + 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.[lower-alpha 5] This notion of the division a(x)/b(x) results in two polynomials, a quotient q(x) and a remainder r(x), such that a = b q + r and degree(r) < degree(b). The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.[15]

When the denominator b(x) is monic and linear, that is, b(x) = x − c for some constant c, then the polynomial remainder theorem asserts that the remainder of the division of a(x) by b(x) is the evaluation f(c).[14] In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.[16]

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.[17] For example, the factored form of

${\displaystyle 5x^{3}-5}$

is

${\displaystyle 5(x-1)\left(x^{2}+x+1\right)}$

over the integers and the reals and

${\displaystyle 5(x-1)\left(x+{\frac {1+i{\sqrt {3}}}{2}}\right)\left(x+{\frac {1-i{\sqrt {3}}}{2}}\right)}$

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 ${\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}}$ with respect to x is the polynomial

$${\displaystyle na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\ldots +2a_{2}x+a_{1}=\sum _{i=1}^{n}a_{i}\cdot i\cdot x^{i-1}.}$$

Similarly, the general antiderivative (or indefinite integral) of ${\displaystyle P}$ is

$${\displaystyle {\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\ldots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}}$$

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 (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ka_k understood to mean the sum of k copies of a_k. For example, over the integers modulo p, the derivative of the polynomial x^p + x is the polynomial 1.[18]

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

• The graph of the zero polynomial

f(x) = 0

is the x-axis.

• The graph of a degree 0 polynomial

f(x) = a₀, where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

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

f(x) = a₀ + a₁x , where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁.

• The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0

is a parabola.

• The graph of a degree 3 polynomial

f(x) = a₀ + a₁x + a₂x² + a₃x³, where a₃ ≠ 0

is a cubic curve.

• The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ , where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

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

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

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

A number a is a root of a polynomial P if and only if the linear polynomial x − a divides P, that is if there is another polynomial Q such that P = (x – a) Q. It may happen that x − a divides P more than once: if (x − a)² divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that (x − a)^m divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots, as, with the above definitions, every number is a root of the zero polynomial, with an undefined multiplicity. With this exception made, the number of roots of P, even counted with their respective multiplicities, cannot exceed the degree of P.[20] The relation between the coefficients of a polynomial and its roots is described by Vieta's formulas.

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

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

When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving is to compute numerical approximations of the solutions.[21] There are many methods for that; some are restricted to polynomials and others may apply to any continuous function. The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1,000 (see Root-finding algorithm).

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

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

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

A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers.[22] The coefficients may be taken as real numbers, for real-valued functions.

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

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

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

A matrix polynomial is a polynomial with square matrices as variables.[23] Given an ordinary, scalar-valued polynomial

${\displaystyle P(x)=\sum _{i=0}^{n}{a_{i}x^{i}}=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}$

this polynomial evaluated at a matrix A is

${\displaystyle P(A)=\sum _{i=0}^{n}{a_{i}A^{i}}=a_{0}I+a_{1}A+a_{2}A^{2}+\cdots +a_{n}A^{n},}$

where I is the identity matrix.[24]

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 M_n(R).

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

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

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

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

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

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

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

${\displaystyle f=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x^{1}+a_{0}x^{0}}$

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

${\displaystyle ax^{k}\;bx^{l}=abx^{k+l}}$

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

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

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

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

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

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[x], it is said that f divides g or f is a divisor of g if there exists a polynomial q in R[x] such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R[x] and r is an element of R such that f(r) = 0, then the polynomial (x − r) divides f. The converse is also true. The quotient can be computed using the polynomial long division.[25][26]

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

${\displaystyle f=q\,g+r}$

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

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 (decimal) 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

${\displaystyle a=r_{m}b^{m}+r_{m-1}b^{m-1}+\dotsb +r_{1}b+r_{0},}$

where m is a nonnegative integer and the r's are integers such that

0 < r_m < b and 0 ≤ r_i < b for i = 0, 1, . . . , m − 1.[27]

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.[28]

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.

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.[29]