For example, consider the quadratic polynomial

For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.

For example, P < sup > SAT </ sup > is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

For example, is a polynomial, but is not, because its second term involves division by the variable x ( 4 / x ), and also because its third term contains an exponent that is not a non-negative integer ( 3 / 2 ).

For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences ; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science ; they are used in calculus and numerical analysis to approximate other functions.

For example, the following is a polynomial:

For example, ( x + 1 )< sup > 3 </ sup > is a polynomial ; its standard form is x < sup > 3 </ sup > + 3x < sup > 2 </ sup > + 3x + 1.

For example, x < sup > 3 </ sup >/ 12 is considered a valid term in a polynomial ( and a polynomial by itself ) because it is equivalent to 1 / 12x < sup > 3 </ sup > and 1 / 12 is just a constant.

For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero.

For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

For instance, the ring ( in fact field ) 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.

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

For some problems, quantum computers offer a polynomial speedup.

The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L < sup > 2 </ sup >, where L is the number of bits in the number to be factored ; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10 < sup > 4 </ sup > qubits without error correction.

For every, there exists a polynomial function p over C such that for all x in, we have, or equivalently, the supremum norm.

For example, the square root of 2 is irrational and not transcendental ( because it is a solution of the polynomial equation x < sup > 2 </ sup > − 2 = 0 ).

( For example if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y-z then we get a total of 3 different polynomials: x + y − z, x + z-y, and y + z − x.

For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.

For example, to compute one prime factor of the natural number N in polynomial time ( no polynomial time factorization algorithm is known in traditional complexity theory ; see integer factorization ):

For it is clear that the total number of ordinary intersections of C and Af must be even ( otherwise, starting in the interior of C, Af could not finally return to the interior ), and the center of rotation at T is the argument of the function, not a value.

For a statement of costs per kilowatt-hour would ignore the fact that many of these costs are not a function of kilowatt-hour output ( or consumption ) of energy.

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist.

: For any set X of nonempty sets, there exists a choice function f defined on X.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

For a reader to assign the title of author upon any written work is to attribute certain standards upon the text which, for Foucault, are working in conjunction with the idea of " the author function ".

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

For small values of m like 1, 2, or 3, the Ackermann function grows relatively slowly with respect to n ( at most exponentially ).

For curves given by the graph of a function, horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound.

For example, for the function

For example the arctangent function satisfies

For example, the function has a horizontal asymptote at y = 0 when x tends both to −∞ and +∞ because, respectively,

For example, the function has

For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.

For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero.

For example, methods to locate a gene within a sequence, predict protein structure and / or function, and cluster protein sequences into families of related sequences.

For each K, the function E < sub > K </ sub >( P ) is required to be an invertible mapping on

For integer order α = n, J < sub > n </ sub > is often defined via a Laurent series for a generating function:

For example, kinship and leadership function both as symbolic systems and as social institutions.