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Term 1 : Unit 3
Binomial Theorem
3.1 The Binomial Expansion of (1 + b) n
3.2 The Binomial Expansion of (a + b) n
Binomial Theorem
3.1 The Binomial Expansion of (1 + b) n
Objectives
n
In this lesson, you will use Pascal’s triangle or   to find the
r
binomial coefficient of any term. You will use the Binomial
Theorem to expand (1 + b)n for positive integer values of n and
identify and find a particular term in the expansion (1+ b)n using the
n r
result, Tr 1   b .
r
Binomial Theorem
aa  b b
a a b
b
2
a 2 ab
a

b


ab
a  b
Separate
Split
A thethe
component
square
in
ofparts.
four.
side
( a + b ).
b2
2
 a 2  2ab  b 2
Binomial
expansion
for n = 2.
Binomial Theorem
ab 2
a 2b
a 2b
3
a
3
 a2 bb
ab
3


a 2b
ab 2
acube
thebcube

A
cuboid
of
And
AASplit
Another
A
small
cuboid
Finally,
a
And
And
with
volume
side
aup
+ as
bof 2
another
cuboid
with
cube
volume
with
cube
another
3ab
2
2
another
2
3
2
3
ashown
volume
a
a ba3a
bb .b  3ab
volume
3
b
3
Binomial Theorem
Pascal’s Triangle
1
1
1
1
1
4
1
2
3
1
Add two adjacent
terms to make the
term below.
3
6
1
4
1
5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1
Now, we will
apply the
triangle to the
binomial
expansion.
Binomial Theorem
Using Pascal’s Triangle to expand (1 + b) 6
1
4
6
4
1
5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1
Take the 6th row of
Pascal’s Triangle.
Use these
numbers as
coefficients.
Form into
a series.
b 
0
b 
b2 
b3 
Write ascending
powers of b from b0 to
b6.
b4 
b5  b 6
Binomial Theorem
Example 1
Expand 1  b  .
5
1  b 
5
 1  5b  10b2  10b3  5b 44  b55
Write ascending powers
of b from b0 to b5.
1
Use these numbers
as coefficients.
5
10 10
5
1
Take the 5th row
of Pascal’s
Triangle.
Binomial Theorem
Use the result 1  b   1  5b  10b 2  10b3  5b 4  b5
5
to find 1  b 
1  b 
5
5
 1   b  
Take care of the
minus signs here.
5
 1  5  b   10  b   10  b   5  b    b 
2
 1  5b  10b2  10b3  5b4  b5
3
4
5
Notice how the
signs alternate
between odd and
even terms.
Binomial Theorem
Use the result 1  b   1  5b  10b 2  10b3  5b 4  b5
5
to find 1  2 x 
1  2 x 
5
5
Remember to include the
coefficients inside the
parentheses.
 1  5  2 x   10  2 x   10  2 x   5  2 x    2 x 
2
3
4
5
 1  5  2 x   10  4 x 2   10 8 x3   5 16 x 4   32 x 5
 1  10 x  40 x 2  80 x3  80 x 4  325
Binomial Theorem
The fifth row of Pascal’s Triangle was
1
5 10 10
5
1
Using Binomial Coefficient notation, these numbers are
 5   5  5   5  5   5
 0   1  2   3  4   5
       
n is the row and r
is the position
(counting from 0).
n
In the expansion of 1  b  the coefficient of b is  
r
n
r
Binomial Theorem
 n  n  n  1 n  2   n  r  1
 r   r r  1 r  2 3  2 1
  
 
n
 0   1,
 
n
 n   1,
 
The binomial
coefficient can be found
from this formula.
 n  n  n  1 n  2 
r 
r!
 
r! – r factorial
 n  r  1
The number of terms in the
numerator and denominator
is always the same.
 5 5 4 3
 8 8 7  6 5 4
 3   3  2 1  10  5   5  4  3  2 1  56
 
 
Binomial Theorem
The Binomial Theorem
1  b 
n
n 0 n 1 n 2 n 3
  b   b   b   b 
0
1
 2
 3
1  b   1  nb 
n
n  n  1
2!
b 
2
n  n  1 n  2 
3!
 n n
 nb
 
b3 
 bn
Binomial Theorem
Example 3
Find the first four terms in the expansion of 1  x
.
2 8
8 2  8  2 2 8 2 3
1  x   1   1   x    2   x    3   x  
 
 
 
8 7 4 8 7  6 6
2
 1  8x 
x 
x 
2 1
3  2 1
 1  8 x 2  28 x 4  56 x 6 
Using this
8
result
Estimate the value of 1.01
2 8
1.01
8

 1   0.1

2 8
 1  8  0.1  28  0.1  56  0.1 
 1  0.08  0.002 8  0.000 056   1.082 856
2
4
6
Binomial Theorem
Example 5
12
 x
Find the terms in x and x in the expansion of 1   .
 2
2
2
33 2
12   x  12 11  x 
 2    2   2 1  4   2 x
Using this

 
 
result
5
5
99 5
12   x  12 1110  9  8  x 
 5    2   5  4  3  2 1   2    4 x



 
12
x
5
3 
Find the coefficient of x in the expansion of  3  2 x  1  
 2
12
x
33 2
99 5

3 
3 
3

2
x
1


3

2
x

x


x


  2  
  2

4

165 5
 99 5 
3  33 2 
  3  x    2x  x  
 
x 
4
 4 
 2 
2
5
Binomial Theorem
Exercise 6.1, qn 3(d), (g)
Find the first four terms, in ascending powers of x in the
expansions of 1  2 x  and 1  2 x
9
1  2 x 
9
.
2 7
98
98 7
2
3
 1 9 2x 
2x 
2x 
2 1
3  2 1
 1  18 x  144 x 2  672 x3 
1  2 x 
2 7
76
7 65
2 2
2 3
 1  7  2 x  
2 x  
2 x  


2 1
3  2 1
2
 1  14 x 2  84 x 4  280 x6 
Binomial Theorem
3.2 The Binomial Expansion of (a + b) n
Objectives
In this lesson, you will use the Binomial Theorem to expand (a + b) n for
positive integer values of n. You will identify and find a particular term in
 n  nr r
n
T

b .
the expansion (a + b) , using the result
.
r 1
 r a
 
Binomial Theorem
The Binomial Theorem
a  b
n
n
  b 
b
n
  a 1     a 1  
 a
  a 
n
2
3

n
n
n
  b    b    b 
n
 a 1                
  1   a   2   a   3   a 
a  b
n
 n  n 1  n  n 2 2  n  n 3 3
 a  a b  a b  a b 
1
 2
 3
n
n
n
  b  
 n a  
    
bn
Binomial Theorem
Example
Expand  a  b  .
5
The combined total
of powers is always
5.
3 2
5
4
a
a

b

a

5
a
b

10
b  10a2 b3  5aab
b 4  b5


5
1
Write
ascending
powers of b
from b0 to
b5.
Write
descending
powers of a
from a5 to
a0.
Use these
numbers as
coefficients.
5
10 10
5
1
Take the 6th
row of
Pascal’s
Triangle.
Binomial Theorem
Example 6(b)
Find, in descending powers of x, the first four terms of
6
1 

x 2  .
x 

Don’t try to simplify
yet – not until the
next stage.
6
2
3
1 

6
5 1 
4 1 
3 1 
 x  2   x  6 x  2   15 x  2   20 x  2  
x 

x 
x 
x 
20
 x  6 x  15  3 
x
6
3
Notice that the
third term is
independent of x.
Binomial Theorem
Example 8(b)
12
1
1 

Find the term in 15 in the expansion of  2 x  2  .
x
x 

r
There is no need
12 
1 
12r 
The general term is    2 x    2  .
to find all the
 x 
terms.
r
1
r 9
For the term in 15 , 12  r   2r  15
x
Looking at
9
12 
1
1 
3
the
The term in 15 is    2 x    2 
combined
9
x
x


 
powers of
x
1
3
 220  8x  18
x
Be careful with
1760
negative
  15
values.
x
Binomial Theorem
Exercise 6.2, qn 6(b)
Find the fourth term in the expansion of
 3x  2 
9

9
6
3
    3x   2  
 3

 84  729 x6   8 
The fourth term is  489888 x6 .
 3x  2 
9
.
There is no need to
find all the terms.
Binomial Theorem
Exercise 6.2, qn 7(c)
10
 3 2
Find the constant term in the expansion of  x  2  .
x 

10  3 10r  2 
The general term is    x    2 
 x 
r
3 10  r   2r  0 r  6
r
10  3 4  2 
0
The term in x is    x    2 
 x 
6
64
12
 210x  12
x
The constant term is 13440.
6
The
combined
powers of x
are 0.