Binomial

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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 except for a monomial.

[edit] 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 with a factor c is obtained by distributing the monomial:43

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 a2 − b2 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)(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 \quad

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

[edit] See also

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