# Binomial

For the scientific naming of living things, see binomial nomenclature.
See binomial (disambiguation) for a list of other meanings.

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

Examples:

• [itex]a + b \quad [itex]
• [itex] {x \over 2} + {x^2 \over 2} [itex]
• [itex] v t - {1 \over 2} g t^2 [itex]

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

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

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

[itex] (a + b)(c + d) = (a + b) c + (a + b) d \ [itex]
[itex] = a c + b c + a d + b d \quad [itex].

The square of a binomial a + b is

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

and the square of the binomial a - b is

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

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

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

A binomial is linear if it is of the form

[itex] a x + b \quad [itex]

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

A complex number is a binomial of the form

[itex] a + i b \quad [itex]

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:

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

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

[itex] (a + b)^n \quad [itex]

can be expanded by means of the binomial theorem or Pascal's triangle. Pascal's triangle is not good to use with large numbers but as a rule of thumb will suffice where the power does not exceed 7.

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