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
x
+
y
{\ displaystyle x + y}
(
x
+
y
)
n
,
n
∈
N
{\ displaystyle (x + y) ^ {n}, \ quad n \ in \ mathbb {N}}
as a polynomial of the -th degree in the variables and .
n
{\ displaystyle n}
x
{\ displaystyle x}
y
{\ displaystyle y}
In algebra , the binomial theorem indicates how to multiply an expression of the form .
(
x
+
y
)
n
{\ 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:
x
{\ displaystyle x}
y
{\ displaystyle y}
n
∈
N
0
{\ displaystyle n \ in \ mathbb {N} _ {0}}
(
x
+
y
)
n
=
∑
k
=
0
n
(
n
k
)
x
n
-
k
y
k
(
1
)
{\ 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 ).
x
{\ displaystyle x}
y
{\ displaystyle y}
0
0
=
1
{\ displaystyle 0 ^ {0} = 1}
The coefficients of this polynomial expression are the binomial coefficients
(
n
k
)
=
n
⋅
(
n
-
1
)
⋯
(
n
-
k
+
1
)
1
⋅
2
⋯
k
=
n
!
(
n
-
k
)
!
⋅
k
!
{\ 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.
n
!
=
1
⋅
2
⋯
n
{\ displaystyle n! = 1 \ cdot 2 \ dotsm n}
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.
(
n
k
)
x
n
-
k
y
k
{\ displaystyle {\ tbinom {n} {k}} x ^ {nk} y ^ {k}}
(
n
k
)
{\ displaystyle {\ tbinom {n} {k}}}
x
n
-
k
y
k
{\ displaystyle x ^ {nk} y ^ {k}}
Z
{\ displaystyle \ mathbb {Z}}
specialization The binomial theorem for the case is called the first binomial formula .
n
=
2
{\ 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.
x
{\ displaystyle x}
y
{\ displaystyle y}
x
⋅
y
=
y
⋅
x
{\ 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:
(
x
+
y
)
n
=
x
n
+
[
∑
k
=
1
n
-
1
(
n
k
)
x
n
-
k
y
k
]
+
y
n
{\ 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.
n
{\ displaystyle n}
n
{\ displaystyle n}
Examples
(
x
+
y
)
3
=
(
3
0
)
x
3
+
(
3
1
)
x
2
y
+
(
3
2
)
x
y
2
+
(
3
3
)
y
3
=
x
3
+
3
x
2
y
+
3
x
y
2
+
y
3
{\ 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}}
(
x
-
y
)
3
=
(
3
0
)
x
3
+
(
3
1
)
x
2
(
-
y
)
+
(
3
2
)
x
(
-
y
)
2
+
(
3
3
)
(
-
y
)
3
=
x
3
-
3
x
2
y
+
3
x
y
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}}
(
a
+
i
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
n
-
k
b
k
i
k
=
∑
k
=
0
,
k
straight
n
(
n
k
)
(
-
1
)
k
2
a
n
-
k
b
k
+
i
∑
k
=
1
,
k
odd
n
(
n
k
)
(
-
1
)
k
-
1
2
a
n
-
k
b
k
{\ 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}}
imaginary unit .
i
{\ 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 .
α
{\ displaystyle \ alpha}
α
{\ displaystyle \ alpha}
The binomial theorem is in its general form:
(
x
+
y
)
α
=
x
α
(
1
+
y
x
)
α
=
x
α
∑
k
=
0
∞
(
α
k
)
(
y
x
)
k
=
∑
k
=
0
∞
(
α
k
)
x
α
-
k
y
k
(
2
)
{\ 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 .
x
,
y
∈
R.
{\ displaystyle x, y \ in \ mathbb {R}}
x
>
0
{\ displaystyle x> 0}
|
y
x
|
<
1
{\ 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.
α
∈
N
{\ displaystyle \ alpha \ in \ mathbb {N}}
x
,
y
∈
C.
{\ displaystyle x, y \ in \ mathbb {C}}
The generalized binomial coefficients used here are defined as
(
α
k
)
=
α
(
α
-
1
)
(
α
-
2
)
⋯
(
α
-
k
+
1
)
k
!
{\ 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.
k
=
0
{\ displaystyle k = 0}
For and , the geometric series results from (2) as a special case .
α
=
-
1
{\ displaystyle \ alpha = -1}
x
=
1
{\ displaystyle x = 1}
literature Individual evidence
↑ Wikibook's evidence archive: Algebra: Rings: Binomial Theorem
Web links
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![(x + y) ^ {n} = x ^ {n} + \ left [\ sum _ {k = 1} ^ {n-1} {\ binom {n} {k}} x ^ {nk} y ^ {k} \ right] + y ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b2f84325523b503c930708198cb281c0742cce)
