Search results
Results from the WOW.Com Content Network
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 ...
6.1 Binomial coefficients as a basis for the space of polynomials. 6.2 Integer-valued polynomials. ... Visualisation of binomial expansion up to the 4th power.
The Gaussian binomial coefficients are defined by: [1] = () (+) () ()where m and r are non-negative integers. If r > m, this evaluates to 0.For r = 0, the value is 1 since both the numerator and denominator are empty products.
The case α = 1 gives the series 1 + x + x 2 + x 3 + ..., where the coefficient of each term of the series is simply 1. The case α = 2 gives the series 1 + 2x + 3x 2 + 4x 3 + ..., which has the counting numbers as coefficients. The case α = 3 gives the series 1 + 3x + 6x 2 + 10x 3 + ..., which has the triangle numbers as coefficients.
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.
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
and the third power has as coefficients the triangular numbers 1, 3, 6, 10, 15, 21, ... Via the binomial theorem expansion, for even , the formula ...
The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [1] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [1]