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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
When a partial fraction term has a single (i.e. unrepeated) binomial in the denominator, the numerator is a residue of the function defined by the input fraction. We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the ...
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.
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
The Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking () = and () =, which gives ( a + b ) n e ( a + b ) x = e ( a + b ) x ∑ k = 0 n ( n k ) a n − k b k , {\displaystyle (a+b)^{n}e^{(a+b)x}=e^{(a+b)x}\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b ...
The first results concerning binomial series for other than positive-integer exponents were given by Sir Isaac Newton in the study of areas enclosed under certain curves. John Wallis built upon this work by considering expressions of the form y = (1 − x 2) m where m is a fraction.
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Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.
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