Binomial

For other uses, see Binomial (disambiguation)}}}.

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

Examples

The product of a binomial with a factor c is obtained by distributing the monomial:

[ c (a + b) = c a + c b \ ]

The product of two binomials a + b and c + d is obtained by distributing twice:

[ (a + b)(c + d) = (a + b) c + (a + b) d \ ]

:::[ = a c + b c + a d + b d \quad ].

The square of a binomial a + b is

[ (a + b)^2 = a^2 + 2 a b + b^2 \quad ]

and the square of the binomial a - b is

[ (a - b)^2 = a^2 - 2 a b + b^2. \quad ]

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

[ a^2 - b^2 = (a + b)(a - b). \quad ]

A binomial is linear if it is of the form

[ a x + b \quad ]

where a and b are constants and x is a variable.

A complex number is a binomial of the form

[ a + i b \quad ]

where i is the square root of minus one.

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

[ a x + b \quad]

[ c x + d \quad ]

[ ----------- \quad]

[ a c x^2 + \ \ \ c b \, x \quad]

::[ \ \ \ \ \ a d x \ \ \ \ \ \, + b d \quad]

[ ----------- \quad ]

[ a c x^2 + (c b + a d) x + b d \quad ]

A binomial a + b raised to the n^th power, represented as

[ (a + b)^n \quad ]

can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle.