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In algebra, a multilinear polynomial [1] is a multivariate polynomial that is linear (meaning affine) in each of its variables separately, but not necessarily simultaneously. It is a polynomial in which no variable occurs to a power of 2 or higher; that is, each monomial is a constant times a product of distinct variables.
Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: ″ + ′ + =, = which is a second-order linear differential equation.
A polynomial expression is an expression built with scalars (elements of K), indeterminates, and the operators of addition, multiplication, ...
The symmetric group S n.This is trivial, as + + + is a generic polynomial for S n.. Cyclic groups C n, where n is not divisible by eight. Lenstra showed that a cyclic group does not have a generic polynomial if n is divisible by eight, and G. W. Smith explicitly constructs such a polynomial in case n is not divisible by eight.
For instance, the polynomial x 2 + 3x + 2 is an example of this type of trinomial with n = 1. The solution a 1 = −2 and a 2 = −1 of the above system gives the trinomial factorization: x 2 + 3x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1). The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.
The trisomy 21 karyotype figure shows the chromosomal arrangement, with the prominent extra chromosome 21. Trisomy 21 is the cause of approximately 95% of observed Down syndrome, with 88% coming from nondisjunction in the maternal gamete and 8% coming from nondisjunction in the paternal gamete. [4]
Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences to create a Newton series that fits the data. The resulting polynomial may be used to extrapolate the data. High-order polynomial extrapolation must be used with due care.
This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function. Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated.