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# binomial

[bahy-noh-mee-uhl]
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:

In algebra, a formula for expansion of the binomial (math.x + math.y) raised to any positive integer power. A simple case is the expansion of (math.x + math.y)2, which is math.x2 + 2math.xmath.y + math.y2. In general, the expression (math.x + math.y)math.n expands to the sum of (math.n + 1)terms in which the power of math.x decreases from math.n to 0 while the power of math.y increases from 0 to math.n in successive terms. The terms can be represented in factorial notation by the expression [math.n!/((math.nmath.r)!math.r!)]math.xmath.nmath.rmath.ymath.r in which math.r takes on integer values from 0 to math.n.

System of naming organisms in which each organism is indicated by two words, the genus (capitalized) and species (lowercase) names, both written in italics. For example, the tea rose is Rosa odorata; the common horse is Equus caballus. The system was developed by Carolus Linnaeus in the mid 18th century. The number of binomial names proliferated as new species were established and more categories were formed, and by the late 19th century the nomenclature of many groups of organisms was confused. International committees in the fields of zoology, botany, bacteriology, and virology have since established rules to clarify the situation. Seealso taxonomy.

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:

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

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

$\left(a x + b\right)\left(c x + d\right) = a c x^2 + \left(a d + b c\right) x + b d.$

A binomial raised to the nth power, represented as

$\left(a + b\right)^n$
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 $a^2+b^2=c^2$.