Fr.: 1) binôme; 2) binomial
1a) An algebraic expression containing 2 terms, as x + y and
2x2 - 3x. In other words, a → polynomial
with 2 terms.
Fr.: coefficient binomial
The factor multiplying the variable in a term of a → binomial expansion. For example, in (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 the binomial coefficients are 1, 4, 6, 4, and 1. In general, the r-th binomial coefficient in the expression (x + y)n is: (n,r) = n!/[r!(n - r)!].
Fr.: binôme différentiel
An expression of the form xm(a + bxn)pdx, where m, n, p, a, and b are constants.
Fr.: distribution binomiale
A probability distribution for independent events for which there are only two possible outcomes i.e., success and failure. The probability of x successes in n trials is: P(x) = [n!/x!(n - x)!] px.qn - x, where p is the probability of success and q = 1 - p the probability of failure on each trial. These probabilities are given in terms of the → binomial theorem expansion of (p + q)n.
Fr.: expansion binomiale
Fr.: nomenclature binomiale
A system introduced by Carl von Linné (1707-1778), the Swedish botanist, in which each organism is identified by two names. The first is the name of the genus (generic name), written with a capital letter. The second is the name of the species (specific name). The generic and specific names are in Latin and are printed in italic type. For example, human beings belong to species Homo sapiens.
Fr.: théorème du binôme
A rule for writing an equivalent expansion of an expression such as (a + b)n without having to perform all multiplications involved. → binomial expansion. The general expression is (a + b)n = &Sigma (n,k)akbn - k, where the summation is from k = 0 to n, and (n,k) = n!/[r!(n - k)!]. For n = 2, (a + b)2 = a2 + 2ab + b2. Historically, the binomial theorem as applied to (a + b)2 was known to Euclid (320 B.C.) and other early Greek mathematicians. In the tenth century the Iranian mathematician Karaji (953-1029) knew the binomial theorem and its accompanying table of → binomial coefficients, now known as → Pascal's triangle. Subsequently Omar Khayyam (1048-1131) asserted that he could find the 4th, 5th, 6th, and higher roots of numbers by a special law which did not depend on geometric figures. Khayyam's treatise concerned with his findings is lost. In China there appeared in 1303 a work containing the binomial coefficients arranged in triangular form. The complete generalization of the binomial theorem for all values of n, including negative integers, was established by Isaac Newton (1642-1727).