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

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 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 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 example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense.

For each one of these polynomials, there should be a best possible choice of coefficients a, b, c, and d that makes the approximation as good as possible.

For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the function in some neighborhood of the point.

For instance,, where 1, and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.

For example, if x < sup > 2 </ sup > − x − 1 = 0, y < sup > 3 </ sup > − y − 1 = 0 and z = xy, then eliminating x and y from z − xy and the polynomials satisfied by x and y using the resultant gives z < sup > 6 </ sup > − 3z < sup > 4 </ sup > − 4z < sup > 3 </ sup > + z < sup > 2 </ sup > + z − 1, which is irreducible, and is the monic polynomial satisfied by the product.

For any field F, the ring of polynomials with coefficients in F is denoted by.

For summations defined in terms of addition of other values than numbers ( such as vectors, matrices, polynomials ), in general of values in some given Abelian group, the value of an empty summation is taken to be the zero element of that group.

For the special case of, there is a closely related set of polynomials, also called the Newton polynomials, that are simply the binomial coefficients for general argument.

* For any two ideals of polynomials I, J, we have

* For any set of polynomials S, let T be the set of their images in A ( X ).

For each set S of polynomials in k ..., x < sub > n </ sub >, define the zero-locus Z ( S ) to be the set of points in A < sup > n </ sup > on which the functions in S simultaneously vanish, that is to say

For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in P < sup > n </ sup > on which the functions in S vanish:

Given a subset V of P < sup > n </ sup >, let I ( V ) be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.

For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent.

For some knot polynomials, such as the Conway, Alexander, and Jones polynomials, the relevant skein relations are sufficient to calculate the polynomial recursively.

For instance, the square pyramidal numbers are given by the Ehrhart polynomials of a square pyramid with an integer unit square as its base and with height one ; the Ehrhart polynomial in this case is ( t + 1 )( t + 2 )( 2t + 3 )/ 6.

For given polynomials and the classical orthogonal polynomials are characterized by being solutions of the differential equation

For this purpose a degree of intellectual and emotional involvement is necessary ; ;

For a moment she thought of answering with the truth but she knew there were men who shied away from virginity, who demanded some degree of education in body as well as mind.

For proper accreditation of schools, teachers in any course must have a degree at least one level above that for which the student is a candidate.

For instance, we cannot know whether even for church members the degree of conformity to Christian standards of morality increased or declined as the proportion of church members in the population rose.

For him, " probability " means a higher chance of occurring, and brings about a higher degree of subjective expectation in the viewer.

For these disciplines, the original master degree abbreviation is combined with the dr. abbreviation thus resulting in for example " dr. ir.

For Derrida the concept of neutrality is suspect and dogmatism is therefore involved in everything to a certain degree.

* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in G. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well.

For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness ; ( providing the system is undamped ).

For non-executive heads of state there is often a degree of censorship by the politically responsible government ( such as the head of government ), discreetly approving agenda and speeches, especially where the constitution ( or customary law ) assumes all political responsibility by granting the crown inviolability ( in fact also imposing political emasculation ) as in the Kingdom of Belgium from its very beginning ; in a monarchy this may even be extended to some degree to other members of the dynasty, especially the heir to the throne.

: For a trivial offence, a free man shall be fined only in proportion to the degree of his offence, and for a serious offence correspondingly, but not so heavily as to deprive him of his livelihood.

For example, a Jones vectors component of () indicates retardation by ( or 90 degree ) compared to 1 ().

For this purpose a writing system is " alphabetic " if it uses symbols to represent individual language sounds, though the degree of correspondence between letters and sounds varies between alphabetic languages.

For other partners, the lead's complete control of the follower, and the follower's relinquishing a greater degree of creative or expressive autonomy is more personally comfortable or satisfying.

For example, Harvard University offers a Master of Liberal Arts degree, which covers biological and social sciences as well as the humanities.

For example, University College Dublin offers the degree, one of the few universities in Europe which does.

For example, Microsoft changed its Office applications from SDI to MDI mode and then back to SDI, although the degree of implementation varies from one component to another.

For instance, Charles Eastman, a man of European and Lakota descent whose father sent both his sons to Dartmouth College, got his medical degree at Boston University and returned to the West to practice.

For thousands of years, mathematicians have attempted to extend their understanding of, sometimes by computing its value to a high degree of accuracy.

For example, in order to become a fully qualified teaching professional in Hong Kong working in a state or government-funded school, one needs to have successfully completed a Postgraduate Diploma in Education (" PGDE ") or a Bachelor's degree in Education (" BEd ") at an approved tertiary educational institution or university.