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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 ...
A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form , where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable.
If is an odd prime and <, then (,) =. Proof: There are exactly two factors of in the numerator of the expression () = ()! / (!), coming from the two terms and in ()!, and also two factors of in the denominator from one copy of the term in each of the two factors of !.
The ensuing dispute is complicated by signs that she may have inherited her father's mental illness and a burgeoning romance. The movie is based on the 2001 play by David Auburn. Raising Genius (2004) – The film is about a boy (Justin Long) who locks himself in the bathroom to work out math equations on the shower wall.
Relationship to the binomial theorem [ edit ] 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 f ( x ) = e a x {\displaystyle f(x)=e^{ax}} and g ( x ) = e b x , {\displaystyle g(x)=e^{bx},} which gives
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively. By comparing coefficients of x r , Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n .
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The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.