Section 14.4 The Binomial Theorem
10.
32
32 3
33 3 3
4444
01 2 3
xxxx
  
 
  
  
11. Applying the Binomial Theorem to
3
3xy
, we have 3, , and 3.axby n 
12.
323
32
33 3 3
3333
01 2 3
xy x xy xy y
  
 
  
  
13. Applying the Binomial Theorem to
3
51x
, we have
5 , 1, and 3.axb n 
14.
332 23
33 3 3
41 4 4 1 4 1 1
01 2 3
xxx x
   
  
   
   
Chapter 14 Sequences, Series, and the Binomial Theorem
15. Applying the Binomial Theorem to
4
21x
, we have
2 , 1, and 4.axb n 
4432
44444
21 2 2 2 2
01234
xxxxx
    
  
    
    
16.
443 2
234
44 4 4 4
31 3 3 1 3 1 31 1
01 2 3 4
xxxxx
    
  
    
    
43 2
4! 4! 4! 4! 4!
81 27 1 9 1 3 1 1
0!40! 1!41! 2!42! 3!43! 4!44!
xx x x    
  
17. Applying the Binomial Theorem to
4
2
2xy
, we have
2
, 2 , and 4.axb y n 


443 2
234
22222
44 4 4 4
22222
01 2 3 4
xy x x y x y xy y
    
 
    
    
Section 14.4 The Binomial Theorem
18.
    
443 2
22222234
44 4 44
01 2 34
xy x xy xy xy y
    
  
    
    
19. Applying the Binomial Theorem to
4
3y
, we have , 3, and 4.ayb n 
4234
43 2
44 4 4 4
3 3333
01 2 3 4
yyyy y
   
  
   
   
20.
4234
43 2
44 4 4 4
44444
01 2 3 4
yyyy y
    
  
    
    
Chapter 14 Sequences, Series, and the Binomial Theorem
21. Applying the Binomial Theorem to
4
3
21x
, we have
3
2 , 1, and 4.axb n 
   


443 2
234
333 3 3
44 4 4 4
21 2 2 1 2 1 2 1 1
01 2 3 4
xxx x x
    
  
    
    
22.
   


443 2
234
555 5 5
44 4 4 4
21 2 2 1 2 1 2 1 1
01 2 3 4
xxx x x
    
  
    
    
20 15 10 5
4! 4! 4! 4! 4!
16 8 4 2
0!40! 1!41! 2!42! 3!43! 4!44!
xx x x

23. Applying the Binomial Theorem to
5
2c
, we have
, 2, and 5.acb n 
5234
54 3 2 5
555555
222222
01 2 3 4 5
cccccc
    
 
    
    
24.
5234
54 3 2 5
55 5 5 5 5
333333
01 2 3 4 5
cccccc
    
 
    
    
Section 14.4 The Binomial Theorem
25. Applying the Binomial Theorem to
5
1x
, we have
, 1, and 5.axb n 
52345
54 3 2
555555
111111
01 2 3 4 5
xxxx x x
    

    
    
26.
52345
54 3 2
555555
222222
01 2 3 4 5
xxxx x x
   
  
   
   
27. Applying the Binomial Theorem to
5
3xy
, we have 3 , , and 5.axb y n 
5
54 3223 4 5
55 5 5 5 5
01 2 3 4 5
3
33 3 3 3
xy
xxyxyxyxy y
   
    
   
   
28.
52345
54 3 2
555555
333333
01 2 3 4 5
xy x x y x y x y xy y
  

  
  
Chapter 14 Sequences, Series, and the Binomial Theorem
29. Applying the Binomial Theorem to
6
2ab
, we have
2 , , and 6.aabb n 
6
30.
623456
65 4 3 2
66 6 6 6 6 6
2222222
01 2 3 4 5 6
ab a ab ab ab ab ab b
     
    
     
     
65 42 33
6! 6! 6! 6!
24 8
0! 6 0 ! 1! 6 1 ! 2! 6 2 ! 3! 6 3 !
6! 6! 6!
aabab ab
 
 
31.
8
2x
80 0 8
88! 8!
First Term ( 0) : 2 1
00! 8 0 !
nr r
n
rabx x
r

 

 
  0! 8!
88
xx
Section 14.4 The Binomial Theorem
32.
8
3x
80 0 8
88! 8!
First Term ( 0) : 3 1
00! 8 0 !
nr r
n
rabx x
r

 

 
  0! 8!
88
xx
33.
10
2xy
0
10 0 10
10 10! 10!
First Term 0 : 2 1
00! 10 0 !
nr r
n
rabxy x
r

  

  
   0!10!
10 10
xx
34.
9
2xy
0
90 9
99! 9!
First Term 0 : 2 1
00! 9 0 !
nr r
n
rabxy x
r

 

 
  0! 9!
99
xx
35.
16
2
1x

16 0
20 32
16 16! 16!
First Term 0 : 1 1
00! 16 0 !
nr r
n
rabx x
r
  

  
   0!16!
32 32
xx
Chapter 14 Sequences, Series, and the Binomial Theorem
36.
17
2
1x

17 0
20 34
17 17! 17!
First Term 0 : 1 1
00! 17 0 !
nr r
n
rabx x
r
  

  
   0! 17!
34 34
xx
37.
20
3
1y

20 0 0
360
20 20! 20!
First Term ( 0) : 1 1
00! 20 0 !
nr r
n
raby y
r
  

  
   0! 20!
60 60
yy
38.
21
3
1y

21 0 0
363
21 21! 21!
First Term 0 : 1 1
00! 21 0 !
nr r
n
raby y
r
  

  
   0! 21!
63 63
yy

39.
6
2xy
40.
2xy
Section 14.4 The Binomial Theorem
41.
9
1x
42.
10
1x
4
10 4 6
10 10! 10 9 8
Fifth Term 4 : 1 1
44! 10 4 !
nr r
n
rabx x
r

   

  
  
76!
432 16!
66
1210xx
44.
8
32
xy
 
85 5
32 910
88! 8 7 6
Sixth Term 5 : 55! 8 5 !
nr r
n
rabxy xy
r
  
  
 
 
5!
5! 32
910 910
56
1xy xy
46.
8
1
2
x



3
83 5
818!18
Fourth Term 3 : 323!83!8
nr r
n
rabx x
r

 


 
 
76 5!
3215!
1
8
55
7xx
48.
10
2xy
6
y
will occur in the seventh term
Chapter 14 Sequences, Series, and the Binomial Theorem
49.
       
443 2213 4
32 3 3 2 3 2 3 2 2
44 4 4 4
01 2 3 4
xx x x x x x x x x

   
 
   
   
50.
     
443 2213 4
23 2 2 3 2 3 2 3 3
44 4 4 4
01 2 3 4
xx x x x x x x x x

    
 
    
    
51.
3
3
11 1 1
33 3 3
xx x x


 


 

 

52.
33
3
22 21
33 33
1
3
3
11
xx xx
xx


 


 

 


 




Section 14.4 The Binomial Theorem
53.
   
323
32 1
33 3 3
13 (1) (1) 3 (1)3 3
01 2 3
iiii
  
  
  
  
54.
   
323
32 1
33 3 3
1 3 ( 1) ( 1) 3 ( 1) 3 3
01 2 3
iiii
  
 
  
  
55.
4
() 7fx x

43 22 344
44
44 4 4 4
77
()7 701 2 3 4
()()
xxhxhxhhx
xh x
fx h fx
hh h
    
 
    
    
 
Chapter 14 Sequences, Series, and the Binomial Theorem
56.
5
() 8fx x
55
54 3223 455
()()()8(8)
55 5 5 5 5
88
01 2 3 4 5
fx h fx x h x
hh
xxhxhxhxhhx
  
    
  
    
    
57. We want to find the (5 1) 6
th
 term.
58. We want to find the (6 1) 7
th
 term.
59.
5
0.28 0.72
60.
5
0.12 0.88
Section 14.4 The Binomial Theorem
68. Exercise 1: 8356nCr
Exercise 2: 7221nCr
69. Graphs
15
and ff are the same. This means that the functions are equivalent. Graphs
24
through ffare increasingly
similar to the graphs of
15
and ff.
71. Applying the Binomial Theorem to
3
1x, we have , 1, and 3.axb n 
323
32
3333
1111
xxxx
  
  
  
72.
4234
43 2
43 2
44 4 4 4
2 2222
01 2 3 4
4! 4! 4! 4! 4!
24816
xxxx x
xx x x
    
  
    
    
 
Chapter 14 Sequences, Series, and the Binomial Theorem
73. Applying the Binomial Theorem to
6
2x, we have , 2, and 6.axb n 
66 5 42 33 24 5 6
66 6 6 6 6 6
2222222
01 2 3 4 5 6
xxxxxxx
     
   
     
     
74. makes sense
75. makes sense
79. true
80. false; Changes to make the statement true will vary. A sample change is: The sum of the binomial coefficients in
n
ab is 2.
n
Section 14.4 The Binomial Theorem
82. Rewrite
33
22
1 as 1xx x x   .
83. In
5
22
xy, the term containing
4
x is the term in which
2
ax is squared. Applying the Binomial Theorem, the
following pattern results. In the first term,
2
x is taken to the fifth power. In the second term,
2
x is taken to the fourth
2
2
10
xy
84.
2
2
2
11213
21223
46
fa a a
aa a
aa
 


85.
2
523
fx x x gx x
 
2
23
23 523
fgx f x
xx

 
Chapter 14 Sequences, Series, and the Binomial Theorem
86.

2
2
1
32224
1
3212
xx
xxx
xx
xxx



Chapter 14 Review Exercises
1. 74
n
an

71 4743
a

2.
2
11
n
n
n
an

1
1
12 3
111 2
a
 
Chapter 14 Review Exercises
3.
1
1!
n
an
1
111
1
11! 0! 1
a

4.
1
1
2
n
nn
a
11 2
11
11
1
a


5.

5222222
2
2 3 21 322 323 324 325 326 3
i
     
6.
4
10111213141
0
1 ! 10!11!12!13!14!
i
i
i


Chapter 14 Sequences, Series, and the Binomial Theorem
9.
1
2
7
7411
a
a

10.
1
2
4
45 9
a
a

  
11.
1
3
2
a
31 21
12.
6
5 6 1 3 5 5 3 5 15 20
a
   
13.
12
81212811282230
a
  