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Binomial
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2 reference results for: Binomial
Columbia Encyclopedia
binomial, polynomial expression (see polynomial) containing two terms, for example, x+y. The binomial theorem, or binomial formula, gives the expansion of the nth power of a binomial (x+y) for n=1, 2, 3, … , as follows:where the ellipsis (…) indicates a continuation of terms following the same pattern. For example, using the formula and reducing fractions, one obtains (x+y)5=x5+5x4y+10x3y2+10x2y3+5xy4+y5. The coefficients 1, n, n (n-1)/1·2, etc., of x and y may also be found from an array known as Pascal's triangle (for Blaise Pascal), formed by adding adjacent numbers to find the number below them as follows:
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Wikipedia

In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. It is the simplest kind of polynomial except for a monomial.

The binomial a^2 - b^2 can be factored as the product of two other binomials:

a^2 - b^2 = (a + b)(a - b).
(This is a special case of the more general formula a^{n+1} - b^{n+1} = (a - b)sum_{k=0}^{n} a^{k},b^{n-k}.)

The product of a pair of linear binomials a x + b and c x + d is:

(a x + b)(c x + d) = a c x^2 + (a d + b c) x + b d.

A binomial a + b raised to the nth power, represented as

(a + b)^n
can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle.

Example

A simple but interesting application of the cited binomial formula is the "(m,n)-formula" for generating Pythagorean triples: for m < n, let a=n^2-m^2, b=2mn, c=n^2+m^2, then a^2+b^2=c^2.

See also

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