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binomial theorem farbin-e donâmin Fr.: théorème du binôme A rule for writing an equivalent expansion of an expression such as
( binomial expansion.
The general expression is
(a + b) = &Sigma (^{n}n,k)a,
where the summation is from ^{k}b^{n - k}k = 0 to n, and
(n,k) = n!/[r!(n - k)!].
For n = 2, (a + b).
Historically, the binomial theorem as applied to (^{2} = a^{2} + 2ab + b^{2}a + b)
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 → ^{2}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). |