# Newton binomial

*binomium of Newton*

The formula for the expansion of an arbitrary positive integral power of a binomial in a polynomial arranged in powers of one of the terms of the binomial:

(*) |

where

are the binomial coefficients. For terms formula (*) takes the form

For an arbitrary exponent , real or even complex, the right-hand side of (*) is, generally speaking, a binomial series.

The gradual mastering of binomial formulas, beginning with the simplest special cases (formulas for the "square" and the "cube of a sum" ) can be traced back to the 11th century. I. Newton's contribution, strictly speaking, lies in the discovery of the binomial series.

#### Comments

The coefficients

are called multinomial coefficients.

**How to Cite This Entry:**

Newton binomial.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Newton_binomial&oldid=13002