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