* Given an algebraic number, there is a unique monic polynomial ( with rational coefficients ) of least degree that has the number as a root.

Given the polynomial

Given a polynomial of degree with zeros < math > z_n < z_

Given a linearly recursive sequence, let C be the transpose of the companion matrix of its characteristic polynomial, that is

Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")

Given a set of n + 1 data points ( x < sub > i </ sub >, y < sub > i </ sub >) where no two x < sub > i </ sub > are the same, one is looking for a polynomial p of degree at most n with the property

Given a square matrix A, we want to find a polynomial whose roots are precisely the eigenvalues of A.

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.

Given n − 1 homogeneous polynomial functions in

Given a quadratic polynomial of the form

Given a quadratic polynomial of the form

Given an integral domain, let be an element of, the polynomial ring with coefficients in.

Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number

Given a multilinear functional M < sub > n </ sub > of degree n ( that is, M < sub > n </ sub > is n-linear ), we can define a polynomial p as

Given a function ƒ on the interval and points in that interval, the interpolation polynomial is that unique polynomial of degree which has value at each point.

Given a vector space V, the polynomials on this space are S ( V *), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing.

Given any decision problem in NP, construct a non-deterministic machine that solves it in polynomial time.

Given a polynomial g, polynomials ( f < sub > 1 </ sub >, ..., f < sub > m </ sub >) and an order on the monomials in k ..., x < sub > n </ sub >, we construct the reduction of g modulo f < sub > 1 </ sub >, ..., f < sub > m </ sub > by the following algorithm.

Fix a base field F and a finite-dimensional vector space V over F. Given a polynomial p ( x ) ∈ F, there is associated to it a companion matrix C whose characteristic polynomial is p ( x ).

* Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions?

Given the above, it is not hard to show that is a polynomial of degree.

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Given a simple, binary component feed, analytical methods such as the McCabe-Thiele method or the Fenske equation can be used.

Given that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age ( i. e. BA ), leads to the equation 19B-8A

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

The functional equation was established by Riemann in his 1859 paper On the Number of Primes Less Than a Given Magnitude and used to construct the analytic continuation in the first place.

Given samples from a population, the equation for the sample skewness above is a biased estimator of the population skewness.

Given the factorization in terms of these matrices, one can now write down immediately an equation

Given the particular differential operators involved, this is a linear partial differential equation.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

Given a simple, binary component feed, analytical methods such as the McCabe-Thiele method or the Fenske equation can be used.

Given the equation

Given the quartic equation

Given a simple, binary component feed, analytical methods such as the McCabe-Thiele method or the Fenske equation can be used.

Given the initial velocity of a particle launched from the ground, the downward ( i. e. gravitational ) acceleration, and the projectile's angle of projection θ ( measured relative to the horizontal ), then a simple rearrangement of the SUVAT equation

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:

Given the complex macro-rheological behavior of blood, it is not surprising that a single equation fails to completely describe the effects of various rheological variables ( e. g., hematocrit, shear rate ).

Given four circles with curvatures k < sub > i </ sub > and centers z < sub > i </ sub > ( for i = 1 ... 4 ), the following equality holds in addition to equation ( 1 ):

Given a Hi of 0. 40, if the H_m is assumed to be 0. 25. then from the equation above RCM count is still high and ANH is not necessary, if BLs does not exceed 2303mL, since the hemotocrit will not fall below H_m.

Given this equation of state, Navier – Stokes and the continuity equation are invariant under the transformations

Given an ordinary non-homogeneous linear differential equation of order n

Given the equation A x = 0 attempt to solve it using