Binomial Theorem
The Binomial Theorem states that for real or complex ,
, and non-negative integer
,
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where is a binomial coefficient. In other words, the coefficients when
is expanded and like terms are collected are the same as the entries in the
th row of Pascal's Triangle.
For example, , with coefficients
,
,
, etc.
Contents
[hide]Proofs
There are a number of different ways to prove the Binomial Theorem, for example by a straightforward application of mathematical induction. The Binomial Theorem also has a nice combinatorial proof:
We can write . Repeatedly using the distributive property, we see that for a term
, we must choose
of the
terms to contribute an
to the term, and then each of the other
terms of the product must contribute a
. Thus, the coefficient of
is the number of ways to choose
objects from a set of size
, or
. Extending this to all possible values of
from
to
, we see that
, as claimed.
Similarly, the coefficients of will be the entries of the
row of Pascal's Triangle. This is explained further in the Counting and Probability textbook [AoPS].
Generalizations
The Binomial Theorem was generalized by Isaac Newton, who used an infinite series to allow for complex exponents: For any real or complex ,
, and
,
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Proof
Consider the function for constants
. It is easy to see that
. Then, we have
. So, the Taylor series for
centered at
is
Usage
Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial , one could factor it as such:
. It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients.