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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 ...
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written (). It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n; this coefficient can be ...
William Betz was active in the movement to reform mathematics in the United States at that time, had written many texts on elementary mathematics topics and had "devoted his life to the improvement of mathematics education". [3] Many students and educators in the US now use the word "FOIL" as a verb meaning "to expand the product of two ...
The usual argument to compute the sum of the binomial series goes as follows. Differentiating term-wise the binomial series within the disk of convergence | x | < 1 and using formula , one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x)u′(x) − αu(x) = 0 with initial condition u(0) = 1.
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.
So, for example, the () = words using 0s and 1s are ,,,,,. To obtain the Gaussian binomial coefficient ( m r ) q {\displaystyle {\tbinom {m}{r}}_{q}} , each word is associated with a factor q d , where d is the number of inversions of the word, where, in this case, an inversion is a pair of positions where the left of the pair holds the letter ...
Argument of a function in mathematical functions; A set of coordinates in a coordinate system; Tuple, a sequence of elements; The greatest common divisor of two numbers; Equivalence class congruence, especially for modular arithmetic or modulo an ideal; A higher order derivative in Lagrange's notation; Binomial or multinomial coefficient
binomial theorem (or binomial expansion) Describes the algebraic expansion of powers of a binomial. bounded function A function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M such that | | for all x in X.