Binomial theorem

The binomial theorem is a theorem of mathematics that, in its simplest form, allows the powers of a binomial x+y , that is, an expression of the form

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

as a polynomial of -th degree in the variables and .

In algebra, the binomial theorem specifies how to $(x+y)^{n}$ multiply out an expression of the form

Binomial theorem for natural exponents

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

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

In particular, this is true for real or complex numbers and (with the convention $0^{0}=1$ ).

The coefficients of this polynomial expression are the binomial coefficients

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

which got their name because of their occurrence in the binomial theorem. With $n!=1\cdot 2\dotsm n$ denotes the factorial of

Comment

The terms ${\tbinom {n}{k}}x^{n-k}y^{k}$ are to be understood as scalar multiplication of the integer ${\tbinom {n}{k}}$ to the ring element $x^{n-k}y^{k}$ That is, here the ring is used in its capacity as a $\mathbb {Z}$ -module is used.

Specialization

The binomial theorem for the case n=2 is called the first binomial formula.

Generalizations

The binomial theorem also holds for elements and in arbitrary unitary rings, provided only these elements commute with each other, i.e. $x\cdot y=y\cdot x$ holds.
Also the existence of the one in the ring is dispensable, provided that one rewrites the theorem into the following form:

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

For more than two summands, there is the multinomial theorem.

Proof

The proof for any natural number can be obtained by complete induction. For any concrete one can also obtain this formula by multiplication out.

Examples

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

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

${\big (}a+ib{\big )}^{n}=\sum \limits _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}i^{k}=\sum _{k=0, \atop k{\text{ gerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k}{2}}a^{n-k}b^{k}+\mathrm {i} \sum _{k=1, \atop k{\text{ ungerade}}}^{n}{\binom {n}{k}}(-1)^{\frac {k-1}{2}}a^{n-k}b^{k}$ , where is the imaginary unit.

Binomial series, theorem for complex exponents

A generalization of the theorem to arbitrary real exponents α $\alpha$ by means of infinite series is due to Isaac Newton. But the same statement is also valid if α is $\alpha$ any complex number.

The binomial theorem in its general form is:

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

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

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

The generalized binomial coefficients used here are defined as

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

In the case k=0 , the result is an empty product whose value is defined as 1.

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

Questions and Answers

Q: What is Binomial expansion?

A: Binomial Expansion is a mathematical method that uses an expression to create a series using the bracket expression (x+y)^n.

Q: What is the basic concept behind Binomial expansion?

A: The basic concept behind Binomial expansion is to expand the power of a binomial expression into a series.

Q: What is a binomial expression?

A: A binomial expression is an algebraic expression containing two terms connected by a plus or minus sign.

Q: What is the formula for Binomial expansion?

A: The formula for Binomial expansion is (x+y)^n, where n is the exponent.

Q: How many types of binomial expansions are there?

A: There are three types of binomial expansions.

Q: What are the three types of binomial expansion?

A: The three types of binomial expansion are - first binomial expansion, second binomial expansion, and third binomial expansion.

Q: How is Binomial expansion useful in mathematical calculations?

A: Binomial expansion is useful in mathematical calculations as it helps to simplify complicated expressions and solve complex problems.