Binomial

From Academic Kids

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:

  • <math>a + b \quad <math>
  • <math> x+3 \quad <math>
  • <math> {x \over 2} + {x^2 \over 2} <math>
  • <math> v t - {1 \over 2} g t^2 <math>

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

<math> c (a + b) = c a + c b \ <math>

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

<math> (a + b)(c + d) = (a + b) c + (a + b) d \ <math>
<math> = a c + b c + a d + b d \quad <math>.

The square of a binomial a + b is

<math> (a + b)^2 = a^2 + 2 a b + b^2 \quad <math>

and the square of the binomial a - b is

<math> (a - b)^2 = a^2 - 2 a b + b^2. \quad <math>

The binomial <math> a^2 - b^2 <math> can be factored as the product of two other binomials:

<math> a^2 - b^2 = (a + b)(a - b). \quad <math>

A binomial is linear if it is of the form

<math> a x + b \quad <math>

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

A complex number is a binomial of the form

<math> a + i b \quad <math>

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:

<math> a x + b \quad<math>
<math> c x + d \quad <math>
<math> ----------- \quad<math>
<math> a c x^2 + \ \ \ c b \, x \quad<math>
<math> \ \ \ \ \ a d x \ \ \ \ \ \, + b d \quad<math>
<math> ----------- \quad <math>
<math> a c x^2 + (c b + a d) x + b d \quad <math>

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

<math> (a + b)^n \quad <math>

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

fr:IdentitÚs remarquables pl:Dwumian

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