Other Definitions
binomial (dict)

Binomial

For the scientific naming of living things, see binomial nomenclature.

In elementary algebra, a binomial is a polynomial with two terms: the sum of two monomials. It is the simplest kind of polynomial. In biology, Binomial nomenclature is a naming convention for all living things. Examples:
  • a + b \quad
  • x+3 \quad
  • {x \over 2} + {x^2 \over 2}
  • v t - {1 \over 2} g t^2
The product of a binomial a + b 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 nth power, represented as
(a + b)^n \quad
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. See also:

 

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