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BINOMIAL THEOREM

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Recognize patterns in binomial expansions.Evaluate a binomial coeicient.

Expan! a binomial raise! to a po"er.#in! a particular term in a binomial expansion.

OBJECTIVES

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Definition: BINOMIAL THEOREM

Patterns in Binomial Exansions

A number of patterns, as follows, begin to appear when we write

the binomial expansion of , where is a positive integer.

and so on.

Definition: BINOMIAL THEOREM

Patterns in Binomial Exansions

A number of patterns, as follows, begin to appear when we write

the binomial expansion of , where is a positive integer.

and so on.

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In each expanded form on the previous slide, the following can be

observed:

1. The first term is , and the exponent on decreases by 1 in

each successive term.

2. The last term is and the exponent on increases by 1 in

each successive term.

. The sum of the exponents on the variables in any term ise!ual to .

". There are terms in the expanded form of .

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n = 0 1

n = 1 1 1

n = 2 1 2 1

n = 3 1 1

n = 4 1 " # " 1

n = 5 1 $ 1% 1% $ 1

n = 6 1 # 1$ 2% 1$ #

1

Definition:

Binomial Coeffi!ients

An interesting pattern for the coefficients in the binomial expansion

can be written in the following triangular arrangement.

This triangular array of coefficients is called the Pascals Triangle.

&hen is small, the use of 'ascal(s triangle is advantageous. )owever, if

is large or a specific term is desired, the use of *inomial Theorem is

more appropriate.

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Definition :

THE BINOMIAL THEOREM

The Binomial T"eorem provides a formula for expanding

expressions of the form , where is a natural number.

+or any binomial and any natural number ,

The specific term of a binomial expansion is

Definition :

THE BINOMIAL THEOREMThe Binomial T"eorem provides a formula for expanding

expressions of the form , where is a natural number.

+or any binomial and any natural number ,

The specific term of a binomial expansion is

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#orm$la for a Binomial Coeffi!ient

In the expansion of , a positive integer, the coefficient of theterm whose variable part is is

#orm$la for t"e t" Term of a Binomial Exansion

The th term of the expansion of is given by

#orm$la for a Binomial Coeffi!ient

In the expansion of , a positive integer, the coefficient of theterm whose variable part is is

#orm$la for t"e t" Term of a Binomial Exansion

The th term of the expansion of is given by

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A. se *inomial Theorem to expand each binomial and

express the result in simplified form.

1.

2.

.

".

E%AMPLES

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*. +ind the term indicated in each expansion.

1. , rdterm

2. , "thterm

. , "thterm

". , #thterm

$. , term that contains

#. , middle term

-. , two middle terms

. , the term that does not contain .

E%AMPLES