Polynomial

Addition and subtraction

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.^[11][17] 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.^[18]

Subtraction of polynomials is similar.

Multiplication

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.^[11] 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.^[18][19]

Composition

Given a polynomial ${\displaystyle f}$ of a single variable and another polynomial ${\displaystyle g}$ 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.^[19] For example, if ${\displaystyle f(x)=x^{2}+2x}$ and ${\displaystyle g(x)=3x+2}$ then $${\displaystyle (f\circ g)(x)=f(g(x))=(3x+2)^{2}+2(3x+2).}$$ 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.^[20]

Division

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.^[21] This is analogous to the fact that the ratio of two integers is a rational number, not necessarily an integer.^[22][23] For example, the fraction ${\displaystyle 1/(x^{2}+1)}$ is not a polynomial, and it cannot be written as a finite sum of powers of the variable ${\displaystyle x}$.

For polynomials in one variable, there is a notion of Euclidean division of polynomials, generalizing the Euclidean division of integers.^[e] This notion of the division ${\displaystyle a(x)/b(x)}$ results in two polynomials, a quotient ${\displaystyle q(x)}$ and a remainder ${\displaystyle r(x)}$, such that ${\displaystyle a=bq+r}$ and ${\displaystyle \deg(r)<\deg(b)}$, where ${\displaystyle \deg(p)}$ is the degree of ${\displaystyle p}$. The quotient and remainder may be computed by any of several algorithms, including polynomial long division and synthetic division.^[24]

When the denominator ${\displaystyle b(x)}$ is monic and linear, that is, ${\displaystyle b(x)=x-c}$ for some constant ${\displaystyle c}$, then the polynomial remainder theorem asserts that the remainder of the division of ${\displaystyle a(x)}$ by ${\displaystyle b(x)}$ is the evaluation ${\displaystyle a(c)}$.^[23] In this case, the quotient may be computed by Ruffini's rule, a special case of synthetic division.^[25]

Factoring

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.^[26] 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.

Calculus

Calculating derivatives and integrals of polynomials is particularly simple, compared to other kinds of functions. The derivative of the polynomial $${\displaystyle P=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots +a_{2}x^{2}+a_{1}x+a_{0}=\sum _{i=0}^{n}a_{i}x^{i}}$$ with respect to ${\displaystyle x}$ is the polynomial $${\displaystyle na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+\dots +2a_{2}x+a_{1}=\sum _{i=1}^{n}ia_{i}x^{i-1}.}$$ Similarly, the general antiderivative (or indefinite integral) of ${\displaystyle P}$ is $${\displaystyle {\frac {a_{n}x^{n+1}}{n+1}}+{\frac {a_{n-1}x^{n}}{n}}+\dots +{\frac {a_{2}x^{3}}{3}}+{\frac {a_{1}x^{2}}{2}}+a_{0}x+c=c+\sum _{i=0}^{n}{\frac {a_{i}x^{i+1}}{i+1}}}$$ where ${\displaystyle c}$ is an arbitrary constant. For example, antiderivatives of ${\displaystyle x^{2}+1}$ have the form ${\displaystyle {\frac {1}{3}}x^{3}+x+c}$.

For polynomials whose coefficients come from more abstract settings (for example, if the coefficients are integers modulo some prime number ${\displaystyle p}$, or elements of an arbitrary ring), the formula for the derivative can still be interpreted formally, with the coefficient ${\displaystyle ka_{k}}$ understood to mean the sum of ${\displaystyle k}$ copies of ${\displaystyle a_{k}}$. For example, over the integers modulo ${\displaystyle p}$, the derivative of the polynomial ${\displaystyle x^{p}+x}$ is the polynomial ${\displaystyle 1}$.^[27]

Graphs

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

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

Solving equations

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(x) = 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 = (x − a) Q. It may happen that a power (greater than 1) 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 (x − a)^m 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,^[29] and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). 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 (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. 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 (1+√5)/2 is the unique positive solution of x² − 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.^[30] 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 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 (see Hilbert's tenth problem). 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.

Trigonometric polynomials

A trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of one or more natural numbers.^[31] 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 the 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 also used in the discrete Fourier transform.

Matrix polynomials

A matrix polynomial is a polynomial with square matrices as variables.^[32] 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.^[33]

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).

Exponential polynomials

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

Rational functions

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

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

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

Laurent polynomials

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

Power series

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

Divisibility

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 ${\displaystyle a\in R,}$ then a is a root of f if and only ${\displaystyle x-a}$ divides f. In this case, the quotient can be computed using the polynomial long division.^[34][35]

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.

Positional notation

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.^[36]

Interpolation and approximation

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.^[37]

In other mathematical fields

Polynomials are frequently used to encode information about some other object. In linear algebra, the characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. In field theory, the minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element.^[38] In algebraic graph theory, the chromatic polynomial of a graph counts the number of proper colourings of that graph.^[39]

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

History of the notation

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, introduced the concept of the graph of a polynomial equation. He popularized the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the as denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.^[40]