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}}$

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

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}$

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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:

${\ displaystyle (x + y) ^ {n} = x ^ {n} + \ left [\ sum _ {k = 1} ^ {n-1} {\ binom {n} {k}} x ^ {nk } y ^ {k} \ right] + y ^ {n}}$.

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

${\ displaystyle (x + y) ^ {3} = {\ binom {3} {0}} \, x ^ {3} + {\ binom {3} {1}} \, x ^ {2} y + { \ binom {3} {2}} \, xy ^ {2} + {\ binom {3} {3}} \, y ^ {3} = x ^ {3} +3 \, x ^ {2} y +3 \, xy ^ {2} + y ^ {3}}$

${\ displaystyle {\ big (} a + ib {\ big)} ^ {n} = \ sum \ limits _ {k = 0} ^ {n} {\ binom {n} {k}} a ^ {nk} b ^ {k} i ^ {k} = \ sum _ {k = 0, \ atop k {\ text {even}}} ^ {n} {\ binom {n} {k}} (- 1) ^ { \ frac {k} {2}} a ^ {nk} b ^ {k} + \ mathrm {i} \ sum _ {k = 1, \ atop k {\ text {odd}}} ^ {n} {\ binom {n} {k}} (- 1) ^ {\ frac {k-1} {2}} a ^ {nk} b ^ {k}}$, 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}$

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}}$