Binomial

The title of this article is ambiguous. For other meanings see binomials .

A binomial ( Latin bi “two”; noun “name”) is a polynomial with two terms in mathematics . More precisely: A binomial is the sum of two monomials . For example are

${\displaystyle a+b,\ x-\pi ,\ x^{2}+y^{2},\ 3ab^{5}-4c^{3},\ {\tfrac {p^{2}}{2}}-q}$

Binomials. The term is not a binomial, but the square of a binomial. ${\displaystyle (a+b)^{2}}$

The term "binomial" goes back to Euclid.

Calculation rules

The following rules apply to the multiplication of two binomials using the associative and distributive law :

${\displaystyle (a+b)(c+d)=ac+ad+bc+bd}$

${\displaystyle (a+b)(c-d)=ac-ad+bc-bd}$

${\displaystyle (a-b)(c-d)=ac-ad-bc+bd}$

In verbal terms: Multiply every term of the first binomial (the first bracket) by every term of the second binomial (the second bracket).

The following special cases are known as binomial formulas :

${\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}$

${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}}$

${\displaystyle (a+b)(a-b)=a^{2}-b^{2}}$

The binomial theorem provides a representation for arbitrarily high powers of a binomial:

${\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}}$

The coefficients are called binomial coefficients and can be defined by this formula. ${\displaystyle {\tbinom {n}{k}}}$

See also

• Monom
• Trinome
• polynomial

Web links

Wiktionary: Binomial  - explanations of meanings, word origins, synonyms, translations

Individual evidence

1. ^ Barth, Federle, Haller: Algebra 1 . Ehrenwirth-Verlag, Munich 1980, p. 187, footnote **, there explanation of the designation binomial formula: "In book X of his elements Euclid names a two-part sum ἐκ δύο ὀνομάτων (ek dýo onomáton), consisting of two names." ${\displaystyle a+b}$