Additional Mathematics
Algebra: Applications
Section 1: The binomial expansion
Notes and Examples
Some new notation is required here. You need to remember that:
n! = n (n  1) (n – 2)……….×1
n!  (n – 1)! = n
and that,
n
Cr 
n!
(n  r )!r !
You do not need to write out expansions of nCr in full if you think first;
6
i.e.
C4 
6 5 4  3  2  1
6!
! gives
4  3  2  1  2 1
4! 2!
65
Only leaves the first two terms of 6! In the numerator.
2 1
When expanding a binomial you need to be very careful, and use brackets
where necessary.
The brackets help to
avoid mistakes with
signs
Example 1
Expand: (2  x) 4
Solution
4  23 (  x ) 4  3  2 2  (  x ) 2 4  3  2  2  (  x ) 3


 (  x) 4
1!
2!
3!
2
3
4
 16  32 x  24 x  8x  x
(2  x) 4  24 
In this case the brackets are a prompt
to raise the ‘3’ to the requisite power
as well as the ‘x’
Example 2
Expand: (1 + 3x) 3
Solution
3  2  (3x) 2 3  2 1 (3 x)3

 (3x) 4
2!
3!
2
3
4
 1  9 x  27 x  27 x  81x
 1  (3  3x) 
© MEI, 04/08/09
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AM Algebra: Applications 1 Notes and Examples
Note: Sometimes we use a full stop as a way of denoting multiplication to
avoid confusion with ‘x’ E.g. 4! = 4. 3. 2. 1. i.e. 3! x = 3. 2. 1. x
You can see further examples using the Flash resources Pascal’s triangle
and binomial coefficients and Finding terms in binomial expansions.
You can test yourself using the interactive questions Evaluating binomial
coefficients and Finding coefficients in binomial expansions.
© MEI, 04/08/09
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