The Binomial Theorem

Expand a power of a binomial using Pascal’s triangle or factorial notation.

Find a specific term of a binomial expansion.

The binomial theorem is used to raise a binomial (a + b) to relatively large powers. To better understand the theorem consider the following powers of (a+b):

baba 1

2222 bababa

3223333 babbaaba

4322344464 babbabaaba

543223455510105 babbababaaba

Note the following patterns for the expansion of

•1. There are n+1 terms, the first and last

•2. The exponents of a decrease and exponents of b increase

•3. The sum of the exponents of a and bin each term is n

nanb

n

a b

Using these patterns the expansion of looks like 8ba

44352678

8

???? babababaa

ba

and the problem now comes down

to finding the value of each coefficient.

This can be done using Pascal’s triangle.

0ba

1ba

2ba

3ba

4ba

5ba 1 5 10 10 5 1

1 4 6 4 1

1 3 3 1

1 2 1

1 1

1

Pascal’s Triangle

The Binomial Theorem Using Pascal’s Triangle

Example• Expand (u v)4.

Solution: We have (a + b)n, where a = u, b = v, and n = 4. We use the 5th row of Pascal’s Triangle:

1 4 6 4 1

Then we have:

4 4 3 1 2 2 1 3 4

4 3 2 2 3 4

1 4 6( ) ( ) 4( ) ( ) ( ) ( ) ( ) 1( ) ( )

4 6 4

u v u u v u v u v v

u u v u v uv v

Another Example• Expand (x 3y)4.

• a = x, b = 3y, and n = 4. We use the 5th row of Pascal’s triangle: 1 4 6 4 1

• Then we have

4 3 1 2 2 3 4

4 3 2 2 3 4

( ) ( ) ( 3 ) ( ) ( 3 ) ( )( 3 ) ( 3 )

12 54 1

1 4 6 4

08 8

1

1

x x y x y x y y

x x y x y xy y

Although Pascal’s triangle can be used to expand a binomial, as the value of the exponent gets larger, it becomes more and more tedious to use this method. The binomial theorem is used for these larger expansions. Before proceeding to the theorem we need some additional notation.

The product of the first n natural numbers is denoted n! and is called n factorial.

nnn 1...321!

and 0! = 1

5!=(1)(2)(3)(4)(5) = 120

The binomial coefficient:let n and r be nonnegative integers with The binomial coefficient is denoted by and

is defined by

nr

r

n

!!

!

rnr

n

r

n

n rC

n rC

or

Evaluate the expression:

3

8

5

8

rn

n

r

n generalIn

6

3

5!2!3!

The Binomial Theorem Using Factorial Notation

Use binomial theorem to find

6542332456

65142

3324156

645762160432048602916729

)2(6

6)2()3(

5

6)2()3(

4

6

)2()3(3

6)2()3(

2

6)2()3(

1

6)3(

0

6

babbabababaa

bbaba

bababaa

6)23( ba 1 6 15 20

156 1

Example

Finding a Specific Term• Finding the (r + 1)st Term

The (r + 1)st term of (a + b)n is

Example: Find the 7th term in the expansion (x2 2y)11.

First, we note that 7 = 6 + 1. Thus, k = 6, a = x2, b= 2y, and n = 11. Then the 7th term of the expansion is

.n r rn

a br

11 6 56 62 2 10 6

11 11!2 or 2 , or 29,568

6 6!5!x y x y x y