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The word polynomial joins two diverse roots: the Greek poly, meaning "many", and the Latin nomen, or "name". It was derived from the term binomial by replacing the Latin root bi-with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. [6]
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. Then, the quotients a i / a n {\displaystyle a_{i}/a_{n}} belong to the field of fractions of R (and possibly are in R itself if a n {\displaystyle a_{n}} happens to be invertible in R ) and the roots r i {\displaystyle r_{i}} are taken in an ...
For n = 1 this results in the already known recurrence formula, just arranged differently, and with n = 2 it forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest m) which implies the evenness or oddness of these polynomials. Three more useful formulas for evaluating Chebyshev ...
The falling factorial occurs in a formula which represents polynomials using the forward difference operator = (+) , which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus
The quadratic formula is valid when the coefficients belong to any field of characteristic different from two, and, in particular, for coefficients in a finite field with an odd number of elements. [9] There are also formulas for roots of cubic and quartic polynomials, which are, in
2.3 Rodrigues formula for Bessel polynomials. 2.4 Associated Bessel polynomials. 3 Zeros. ... In mathematics, the Bessel polynomials are an orthogonal sequence of ...
Faulhaber's formula is also called Bernoulli's formula. Faulhaber did not know the properties of the coefficients later discovered by Bernoulli. Rather, he knew at least the first 17 cases, as well as the existence of the Faulhaber polynomials for odd powers described below. [2] Jakob Bernoulli's Summae Potestatum, Ars Conjectandi, 1713
This allows formulas to be given for the n th cyclotomic polynomial when n has at most one odd prime factor: If p is an odd prime number, and h and k are positive integers, then Φ 2 m ( x ) = x 2 m − 1 + 1 , {\displaystyle \Phi _{2^{m}}(x)=x^{2^{m-1}}+1\;,}