The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.

1. 9.6 Binomial Theorem
Chapter 9 Sequences, Probability, and Counting
Theory

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Apply the Binomial Theorem.

3. Binomial Series
⚫ A binomial squared becomes
⚫ A binomial cubed becomes
( )
+ = + +
2 2 2
2
a b a ab b
( ) ( )( )
+ = + +
3 2
a b a b a b
( )( )
= + + +
2 2
2
a b a ab b
= + + + + +
2 2
3 3
2 2
2 2
a b a b ab
a b b
a
= + + +
3 2 2 3
3 3
a a b ab b

4. Binomial Series (cont.)
⚫ As you may recall from Algebra II, the coefficients
correspond to rows from Pascal’s Triangle
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1

5. Binomial Series (cont.)
⚫ Example: Expand ( )
+
5
2 1
x

6. Binomial Series (cont.)
⚫ Example: Expand
a = 2x and b = 1; the exponents begin and end at 5
(a goes down while b goes up). Looking at row 5 on the
triangle, our coefficients are 1, 5, 10, 10, 5, 1, so we write
our expression as follows:
(Notice that the exponents apply to the entire term of the
binomial, not just the variable.)
( )
+
5
2 1
x
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
+ + + + +
5 4 3 2 2 3 4 5
5 10 1
1 1 1
2 2 2 1
2 1
5 2
0
x x x x x
= + + + + +
5 4 3 2
32 80 80 40 10 1
x x x x x
+ + + + +
5 4 3 2 2 3 4 5
5 10 10 5
a a a a a
b b b b b

7. Binomial Series (cont.)
⚫ Consider the binomial series :
If we multiply the coefficient of a term by a fraction
consisting of the exponent of a over the term number,
we get the coefficient of the next number.
( )
+
7
a b
= + + + + + + +
7 6 5 2 4 3 3 4 2 5 6 7
7 21 35 35 21 7
a a b a b a b a b a b ab b
8
7
6
5
4
3
2
1
   
= =
   
   
exp. 7
coeff. 1 7,
term # 1
 
=
 
 
6
7 21,
2
 
 
 
5
21 =35, ...
3

8. Binomial Series (cont.)
⚫ Now let’s see what happens to if we don’t
simplify the fractions as we calculate them:
( )
+
8
a b
1
2
3
4
5
8
a
7
8
1
a b
6 2
8 7
1 2
a b
5 3
8 7 6
1 2 3
a b
4 4
8 7 6 5
1 2 3 4
a b
Do you see the pattern?
What is it?

9. Binomial Series (cont.)
⚫ The coefficients of a binomial series can be written as
factorials. For example, let’s look at the coefficient for
the fourth term:
=
8 7 6 8 7 6
1 2 3 1 2 3
=
8 7 6 5!
1 2 3 5!
=
8!
3! 5!

10. Binomial Series (cont.)
⚫ Looking back at the original expression:
Notice how the numbers in the coefficient expression
are found elsewhere in the expression.
⚫ 8 is the value of the exponent to which (a + b) is
raised.
⚫ 5 is the value of a’s exponent and 3 is the value of b’s.
⚫ The exponent of b is always one less than the term
number (4).
( )
+ = + +
5 3
8 !
... ...
! !
5
3
8
a b a b

11. Binomial Theorem
⚫ The formula for the term containing br of (a + b)n,
therefore, is
or nCr
⚫ Example: Find the term containing y6 of
( )
−
−
!
! !
n r r
n
a b
r n r
n
r
 
=  
 
( )
10
8
x y
−

12. Binomial Theorem (cont.)
⚫ The formula for the term containing br of (a + b)n,
therefore, is
or nCr
⚫ Example: Find the term containing y6 of
( )
−
−
!
! !
n r r
n
a b
r n r
n
r
 
=  
 
( )
10
8
x y
−
( )( ) ( ) ( )
6
10 6 4 6
10
8 210 262144
6
x y x y
−
 
− =
 
  4 6
55,050,240x y
=

13. Binomial Theorem (cont.)
⚫ Example: Find the term in which contains f 9.
( )
15
e f
−

14. Binomial Theorem (cont.)
⚫ Example: Find the term in which contains f 9.
Since n = 15, n – r = 15 – 9 = 6. Therefore the term is
(When dealing with negative terms such as f, recall that
even exponents will produce positive terms and odd
exponents will produce negative terms.)
( )
15
e f
−
( )
9
6 6 9
15
5005
6
e f e f
 
− = −
 
 

15. Binomial Theorem (cont.)
⚫ Similarly, the kth term of binomial expansion of
is found by realizing that the exponent of b will be k – 1,
which gives us the formula:
(replace r with k – 1)
( )
n
a b
+
( )
1 1
1
n k k
n
a b
k
− − −
 
 
−
 

16. Binomial Theorem (cont.)
⚫ Example: Find the 4th term of ( )
12
2c d
−

17. Binomial Theorem (cont.)
⚫ Example: Find the 4th term of
n = 12, k = 4, which means that k – 1 = 3
( )
12
2c d
−
( ) ( ) ( )( )
9 3 9 3
12
2 220 512
3
c d c d
 
− = −
 
 
9 3
112,640c d
= −

18. Binomial Theorem
⚫ Your calculator can also find the coefficient:
4th term of
n = 12, k – 1 = 3, n – (k –1) = 9
2l9¢b5312,3·
( )
12
2c d
−

19. Binomial Theorem (cont.)
⚫ Desmos can also find the coefficient using a function
called nCr(n, r):
4th term of
n = 12, k – 1 = 3, n – (k –1) = 9
( )
12
2c d
−
( ) ( )
9 3
12
2
3
c d
 
−
 
 

20. Binomial Theorem Practice
⚫ Example: Expand ( )
−
5
2 3
x

21. Binomial Theorem Practice
⚫ Example: Expand ( )
−
5
2 3
x
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
5 0 4 1 3 2
2 3 1 4 0 5
5 5 5
2 3 2 3 2 3
0 1 2
5 5 5
2 3 2 3 2 3
3 4 5
x x x
x x x
     
= − + − + −
     
     
     
+ − + − + −
     
     

22. Binomial Theorem Practice
⚫ Example: Expand ( )
−
5
2 3
x
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
5 0 4 1 3 2
2 3 1 4 0 5
5 5 5
2 3 2 3 2 3
0 1 2
5 5 5
2 3 2 3 2 3
3 4 5
x x x
x x x
     
= − + − + −
     
     
     
+ − + − + −
     
     
( )( )( ) ( )( )( ) ( )( )( )
( )( )( ) ( )( )( ) ( )( )( )
5 4 3
2
1 32 1 5 16 3 10 8 9
10 4 27 5 2 81 1 1 243
x x x
x x
= + − +
+ − + + −

23. Binomial Theorem Practice
⚫ Example: Expand ( )
−
5
2 3
x
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
5 0 4 1 3 2
2 3 1 4 0 5
5 5 5
2 3 2 3 2 3
0 1 2
5 5 5
2 3 2 3 2 3
3 4 5
x x x
x x x
     
= − + − + −
     
     
     
+ − + − + −
     
     
( )( )( ) ( )( )( ) ( )( )( )
( )( )( ) ( )( )( ) ( )( )( )
5 4 3
2
1 32 1 5 16 3 10 8 9
10 4 27 5 2 81 1 1 243
x x x
x x
= + − +
+ − + + −
5 4 3 2
32 240 720 1080 810 243
x x x x x
= − + − + −

24. Classwork
⚫ College Algebra 2e
⚫ 9.6: 8-36 (4); 9.5: 36-52 (4); 9.4: 46-62 (even)
⚫ 9.6 Classwork Check
⚫ Quiz 9.5