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There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality. In geometric terms, the feasible region defined by all values of x {\displaystyle \mathbf {x} } such that A x ≤ b {\textstyle A\mathbf {x} \leq \mathbf {b} } and ∀ i , x i ≥ 0 ...
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of . In particular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariate polynomials. For example, the polynomial
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.
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.
Polynomial transformations have been applied to the simplification of polynomial equations for solution, where possible, by radicals. Descartes introduced the transformation of a polynomial of degree d which eliminates the term of degree d − 1 by a translation of the roots. Such a polynomial is termed depressed. This already suffices to solve ...
If one root r of a polynomial P(x) of degree n is known then polynomial long division can be used to factor P(x) into the form (x − r)Q(x) where Q(x) is a polynomial of degree n − 1. Q ( x ) is simply the quotient obtained from the division process; since r is known to be a root of P ( x ), it is known that the remainder must be zero.
The concept of parent function is less clear or inapplicable polynomials of higher degree because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x n, or, to simplify further, x 2 when n is even and x 3 for odd n. Turning points may be established by ...