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In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
Note: "lc" stands for the leading coefficient, the coefficient of the highest degree of the variable. This algorithm computes not only the greatest common divisor (the last non zero r i ), but also all the subresultant polynomials: The remainder r i is the (deg( r i −1 ) − 1) -th subresultant polynomial.
Since ! is the product of the integers 1 through n, we obtain at least one factor of p in ! for each multiple of p in {,, …,}, of which there are ⌊ ⌋.Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for (!
It may happen that this makes the coefficient 0. [12] Polynomials can be classified by the number of terms with nonzero coefficients, so that a one-term polynomial is called a monomial, [d] a two-term polynomial is called a binomial, and a three-term polynomial is called a trinomial. A real polynomial is a polynomial with real coefficients.
Every univariate polynomial with real coefficients of degree larger than two has a factor of degree two with real coefficients. Every univariate polynomial with real coefficients of positive degree can be factored as c p 1 ⋯ p k , {\displaystyle cp_{1}\cdots p_{k},} where c is a real number and each p i {\displaystyle p_{i}} is a monic ...
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
Another abbreviated method is polynomial short division (Blomqvist's method). Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainder R such that A = BQ + R,
The following rules apply to the coefficients of all adjacent pairs of terms of the trinomial expansion: The exponent of one of the variables remains unchanged (B in this case) and can be ignored. For the other two variables, one exponent increases by 1 and one exponent decreases by 1. The exponents of A are 3 and 2 (the larger being in the ...
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