The Binomial Expansion
CURRICULUM CONTENT
The Binomial Expansion
In this presentation we will develop a formula to
enable us to find the terms of the expansion of
n
(a b )
where n is any positive integer.
We call the expansion binomial as the original
expression has 2 parts.
The Binomial Expansion
We know that
( a b ) ( a b )( a b )
2
a 2 2 ab b 2
We can write this as
2
1
a 2 ab 1
so the coefficients of the terms are 1, 2 and 1
The Binomial Expansion
( a b ) ( a b )( a b )
3
( a b )( a 2 2 ab b 2 )
1a 3 2 a 2 b 1 ab 2
2
2 1 3
1
2
a b ab b
1a
3a b 3 ab
2
The Binomial Expansion
( a b ) ( a b )( a b )
3
( a b )( a 2 2 ab b 2 )
1a 3 2 a 2 b 1 ab 2
2
2 1 3
1
2
a b ab b
1 a 3 3 a 2 b 3 ab 2 1 b 3
so the coefficients of the expansion of
are 1, 3, 3 and 1
(a b)
The Binomial Expansion
So, we now have
Expression
(a b )
(a b )
(a b )
Coefficients
1
1
1
2
3
1
3
1
4
The Binomial Expansion
So, we now have
Expression
(a b )
(a b )
(a b )
Coefficients
1
1
1
2
3
1
3
1
4
Each number in a row can be found by adding the 2
coefficients above it.
The Binomial Expansion
So, we now have
Expression
(a b )
(a b )
(a b )
Coefficients
1
1
1
2
3
1
3
1
4
Each number in a row can be found by adding the 2
coefficients above it.
The 1st and last numbers are always 1.
The Binomial Expansion
So, we now have
Expression
0
Coefficients
(a b )
(a b)1
(a b )
(a b )
(a b )
1
1
1
1
1
2
3
4
1
3
1
4
To make a triangle of coefficients, we can fill in
the obvious ones at the top.
The Binomial Expansion
The triangle of binomial coefficients is called
Pascals triangle, after the French mathematician.
Notice that the 4th row gives the coefficients of
( a b )3
. . . but its easy to know which row we want as,
for example,
( a b ) 3 starts with
1 3
(a b )
1 10 . . .
10
will start
. . .
The Binomial Expansion
Exercise
Find the coefficients in the expansion of
Solution: We need 7 rows
1
1
1
1
1
Coefficients
1
2
3
4
(a b ) 6
3
6
10
15
1
1
4
10
20
1
5
15
The Binomial Expansion
We usually want to know the complete expansion not
just the coefficients.
5
e.g. Find the expansion of
(a b )
Solution: Pascals triangle gives the coefficients
10
10
The full expansion is
1a
b 5 a b 10 a b 10 a b
5 0
4 1
3 2
2 3
ab
Tip: The powers in each term sum to 5
The Binomial Expansion
Example 1
1. Find the expansion of
powers of x.
Solution:
The coefficients are
1
So,
(1 2 x ) 5 in ascending
10
10
(1 2 x ) 5
1 5 ( 2 x ) 10( 2 x ) 2 10( 2 x ) 3 5 ( 2 x ) 4 ( 2 x ) 5
1 10 x 40 x 80 x 80 x 32 x
2
The Binomial Expansion
EXERCISE 1
The Binomial Expansion
If we want the first few terms of the expansion
of, for example, ( a b ) 20 , Pascals triangle is not
helpful.
We will now develop a method of getting the
coefficients without needing the triangle.
The Binomial Expansion
Generalizations Binomial Theorem
The binomial expansion of ( a b ) n in ascending
powers is given by
( a b)
n
n 1
n2 2
C0 a b C1 a b C2 a b . . . b
n
Specific Term of an Expansion
If only a specific term of an expansion is required,
the binomial theorem can be used to determine such a
term without computing all the rows of Pascals
triangle or all the preceding coefficients.
U r 1 Cr a
n
n r r
where r is any integer from 0 to n.
The Binomial Expansion
e.g.2 Find the first 4 terms in the expansion of
(1 x ) 18 in ascending powers of x.
Solution:
( 1 x)
C0 (1) ( x)
18 18
18
18
C 1 ( x )
18
18
C2 ( x)
C 3 ( x ) . . .
1 18 x 153 x 2 816 x 3 . . .
3
The Binomial Expansion
EXERCISE 2
The Binomial Expansion
Powers of a + b
e.g.4 Find the 5th term of the expansion of
in ascending powers of x.
Solution: The 5th term contains
It is
U r 1 C r a
n
n r r
U r 1 C4 (2) x
12
(2 x )
495 (2) x
4
126720 x
8
These numbers
will always be
the same.
12
The Binomial Expansion
EXERCISE 3
The Binomial Expansion
SUMMARY
The binomial expansion of
powers of x is given by
n
n in ascending
(a b )
(a b )
n
C 0 a n n C 1 a n 1 b n C 2 a n 2 b 2 . . . b n
The ( r + 1 ) th term is
C r a nr b r
The expansion of
(1 x) n is
(1 x ) n n C 0
C1 x
C2 x2 . . . xn
The Binomial Expansion
Exercise
1. Find the 1st 4 terms of the expansion of ( 2 3 x ) 8
in ascending powers of x.
Solution:
8
C 0 28
C 1 2 7 (3 x )
C 2 2 6 (3 x ) 2
C 3 2 5 (3 x ) 3
256 3072 x 16128 x 2 48384 x 3
2. Find the 6th term of the expansion of
in ascending powers of x.
Solution:
13
C 5 ( x )
1287 x 5
(1 x ) 1 3
The Binomial Expansion
