Binomial theorem

The binomial theorem is a theorem of mathematics that makes it in its simplest form, the powers of a binomial, that is an expression of the form

As a polynomial of degree n in the variables x and y express.

In algebra specifies the binomial theorem, as an expression of the form is auszumultiplizieren.

Binomial theorem for natural exponents

For all elements of a commutative unitary ring and for all natural numbers, the equation is:

This is especially true for real or complex numbers and. (Note this. )

The coefficients of this polynomial are the binomial coefficients

Have received its name because of their occurrence in the binomial theorem. With this, the Faculty of is denoted.

Remark

The terms are to be understood as scalar multiplication of whole numbers to the ring element, ie here the ring is used in its capacity as a module.

Specialization

The binomial theorem for the case is, first binomial formula.

Generalizations

  • The binomial theorem is also true for elements and in any unitary rings, provided only that these elements commute, ie applies.
  • Even the existence of the one in the ring is dispensable, if one rewrites the theorem in the following form:
  • For more than two summands, there is the Multinomialtheorem.

Derivation

The evidence for any natural number can be provided by complete induction. For each concrete can be obtained this formula by multiplying out.

Example

Binomial series, theorem for complex exponents

A generalization of the theorem to arbitrary real exponent α by means of infinite series is due to Isaac Newton. This same statement is also valid if α is an arbitrary complex number.

The binomial theorem is in its general form:

This series is called binomial series converges for all with and.

In the special case is equation (2) in (1 ) above and is then valid for all even since the series then terminates.

As used herein the generalized binomial coefficients are defined as

In the case of k = 0, a blank product, the value is defined as 1 is formed.

For α = -1 and x = 1 follows from ( 2) as a special case of the geometric series.

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