Binomial theorem: Difference between revisions
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In [[elementary algebra]], the '''binomial theorem''' is the identity that states that for any non-negative [[integer]] ''n'', | In [[elementary algebra]], the '''binomial theorem''' or the binomial expansion is a mechanism by which expressions of the form <math>(x + y)^n</math> can be expanded. It is the identity that states that for any non-negative [[integer]] ''n'', | ||
: <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math> | : <math> (x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}, </math> | ||
where | |||
: <math> {n \choose k} = \frac{n!}{k!(n - k)!} </math> | |||
is a [[binomial coefficient]]. Another useful way of stating it is the following: | |||
<math>(x + y)^n = {n \choose 0} x^n + {n \choose 1} x^{n-1} y + {n \choose 2} x^{n-2} y^2 + \ldots + {n \choose n} y^n</math> | |||
===Pascal's triangle=== | |||
An alternate way to find the binomial coefficients is by using [[Pascal's triange]]. The triangle is built from apex down, starting with the number one alone on a row. Each number is equal to the sum of the two numbers directly above it. | |||
n=0 1 | |||
n=1 1 1 | |||
n=2 1 2 1 | |||
n=3 1 3 3 1 | |||
n=4 1 4 6 4 1 | |||
n=5 1 5 10 10 5 1 | |||
Thus, the binomial coefficients for the expression <math>(x + y)^4</math> are 1, 3, 6, 4, and 1. | |||
==Proof== | |||
One way to prove this identity is by [[mathematical induction]]. | One way to prove this identity is by [[mathematical induction]]. | ||
'''Base case''': n = 0 | '''Base case''': n = 0 | ||
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and the proof is complete. | and the proof is complete. | ||
== | == Examples == | ||
These are the expansions from 0 to 6. | |||
<math> \begin{align} | <math> \begin{align} |
Revision as of 07:34, 9 August 2010
In elementary algebra, the binomial theorem or the binomial expansion is a mechanism by which expressions of the form can be expanded. It is the identity that states that for any non-negative integer n,
where
is a binomial coefficient. Another useful way of stating it is the following:
Pascal's triangle
An alternate way to find the binomial coefficients is by using Pascal's triange. The triangle is built from apex down, starting with the number one alone on a row. Each number is equal to the sum of the two numbers directly above it.
n=0 1 n=1 1 1 n=2 1 2 1 n=3 1 3 3 1 n=4 1 4 6 4 1 n=5 1 5 10 10 5 1
Thus, the binomial coefficients for the expression are 1, 3, 6, 4, and 1.
Proof
One way to prove this identity is by mathematical induction.
Base case: n = 0
Induction case: Now suppose that it is true for n : and prove it for n + 1.
and the proof is complete.
Examples
These are the expansions from 0 to 6.
Newton's binomial theorem
There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)n as an infinite series when n is not an integer or is not positive.