# Binomial theorem

The binomial theorem is a proposition of mathematics which , in its simplest form, allows the powers of a binomial , i.e. an expression of the form ${\ displaystyle x + y}$

${\ displaystyle (x + y) ^ {n}, \ quad n \ in \ mathbb {N}}$

as a polynomial of the -th degree in the variables and . ${\ displaystyle n}$${\ displaystyle x}$${\ displaystyle y}$

In algebra , the binomial theorem indicates how to multiply an expression of the form . ${\ displaystyle (x + y) ^ {n}}$

## Binomial theorem for natural exponents

For all elements and a commutative unitary ring and for all natural numbers the equation applies: ${\ displaystyle x}$${\ displaystyle y}$ ${\ displaystyle n \ in \ mathbb {N} _ {0}}$

${\ displaystyle (x + y) ^ {n} = \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} x ^ {nk} y ^ {k} \ quad (1) }$

In particular, this is true for real or complex numbers and (by convention ). ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle 0 ^ {0} = 1}$

The coefficients of this polynomial expression are the binomial coefficients

${\ displaystyle {\ binom {n} {k}} = {\ frac {n \ cdot (n-1) \ dotsm (n-k + 1)} {1 \ cdot 2 \ dotsm k}} = {\ frac {n!} {(nk)! \ cdot {k!}}}}$,

which got their name because of their appearance in the binomial theorem. With which this Faculty of designated. ${\ displaystyle n! = 1 \ cdot 2 \ dotsm n}$${\ displaystyle n}$

### comment

The terms are to be understood as a scalar multiplication of the whole number to the ring element , i.e. H. Here, the ring than in his capacity - module used. ${\ displaystyle {\ tbinom {n} {k}} x ^ {nk} y ^ {k}}$${\ displaystyle {\ tbinom {n} {k}}}$${\ displaystyle x ^ {nk} y ^ {k}}$${\ displaystyle \ mathbb {Z}}$

### specialization

The binomial theorem for the case is called the first binomial formula . ${\ displaystyle n = 2}$

### Generalizations

• The binomial theorem also applies to elements and in arbitrary unitary rings , provided that only these elements commute with one another, i.e. H. applies.${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle x \ cdot y = y \ cdot x}$
• The existence of the one in the ring is also dispensable, provided that the proposition is paraphrased in the following form:
${\ displaystyle (x + y) ^ {n} = x ^ {n} + \ left [\ sum _ {k = 1} ^ {n-1} {\ binom {n} {k}} x ^ {nk } y ^ {k} \ right] + y ^ {n}}$.

### proof

The proof for any natural number can be given by complete induction . This formula can also be obtained for each concrete by multiplying it out. ${\ displaystyle n}$${\ displaystyle n}$

### Examples

${\ displaystyle (x + y) ^ {3} = {\ binom {3} {0}} \, x ^ {3} + {\ binom {3} {1}} \, x ^ {2} y + { \ binom {3} {2}} \, xy ^ {2} + {\ binom {3} {3}} \, y ^ {3} = x ^ {3} +3 \, x ^ {2} y +3 \, xy ^ {2} + y ^ {3}}$
${\ displaystyle (xy) ^ {3} = {\ binom {3} {0}} \, x ^ {3} + {\ binom {3} {1}} \, x ^ {2} (- y) + {\ binom {3} {2}} \, x (-y) ^ {2} + {\ binom {3} {3}} \, (- y) ^ {3} = x ^ {3} - 3 \, x ^ {2} y + 3 \, xy ^ {2} -y ^ {3}}$
${\ displaystyle {\ big (} a + ib {\ big)} ^ {n} = \ sum \ limits _ {k = 0} ^ {n} {\ binom {n} {k}} a ^ {nk} b ^ {k} i ^ {k} = \ sum _ {k = 0, \ atop k {\ text {even}}} ^ {n} {\ binom {n} {k}} (- 1) ^ { \ frac {k} {2}} a ^ {nk} b ^ {k} + \ mathrm {i} \ sum _ {k = 1, \ atop k {\ text {odd}}} ^ {n} {\ binom {n} {k}} (- 1) ^ {\ frac {k-1} {2}} a ^ {nk} b ^ {k}}$, where is the imaginary unit .${\ displaystyle i}$

## Binomial series, theorem for complex exponents

A generalization of the theorem to any real exponent by means of infinite series is thanks to Isaac Newton . But the same statement is also valid if is any complex number . ${\ displaystyle \ alpha}$${\ displaystyle \ alpha}$

The binomial theorem is in its general form:

${\ displaystyle (x + y) ^ {\ alpha} = x ^ {\ alpha} \ left (1 + {\ tfrac {y} {x}} \ right) ^ {\ alpha} = x ^ {\ alpha} \ sum _ {k = 0} ^ {\ infty} {\ binom {\ alpha} {k}} \ left ({\ frac {y} {x}} \ right) ^ {k} = \ sum _ {k = 0} ^ {\ infty} {\ binom {\ alpha} {k}} x ^ {\ alpha -k} y ^ {k} \ quad (2)}$.

This series is called the binomial series and converges for all with and . ${\ displaystyle x, y \ in \ mathbb {R}}$${\ displaystyle x> 0}$${\ displaystyle \ left | {\ tfrac {y} {x}} \ right | <1}$

In the special case , equation (2) changes into (1) and is then even valid for all , since the series then breaks off. ${\ displaystyle \ alpha \ in \ mathbb {N}}$${\ displaystyle x, y \ in \ mathbb {C}}$

The generalized binomial coefficients used here are defined as

${\ displaystyle {\ binom {\ alpha} {k}} = {\ frac {\ alpha (\ alpha -1) (\ alpha -2) \ dotsm (\ alpha -k + 1)} {k!}}}$

In this case , an empty product is created, the value of which is defined as 1. ${\ displaystyle k = 0}$

For and , the geometric series results from (2) as a special case . ${\ displaystyle \ alpha = -1}$${\ displaystyle x = 1}$