Binomial Theorem
The Binomial Theorem is used to expand out brackets of the form (a  b)n , where n is a whole number.
n
Coefficients
( a  b) n
0
( a  b)  1
1
2
(a  b)1  a  b
3
(a  b)3  a3  3a 2b  3ab 2  b3
4
(a  b)  a  4a b  6a b  4ab  b
1
0
(a  b)  a  2ab  b
2
2
4
4
1 1
1 2 1
2
3
2 2
1 3 3 1
3
1 4 6 4 1
4
Note 1: The coefficients in these expansions form Pascal’s Triangle. These numbers can also be found
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using the nCr button on a calculator. For example, the coefficients for the expansion of  a  b  are:
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C0  1
7
C1  7
7
C4  35
7
C5  21
7
C2  21
7
C6  7
7
7
C3  35
C7  1
Note 2: As the power of a decreases by 1, the power of b increases by 1. In each term, when you add
together the powers of a and b together you get n.
So,
a  b
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 a 7  7a 6b  21a 5b 2  35a 4b3  35a 3b 4  21a 2b5  7ab 6  b 7
Example 1:
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Find the expansion of  3x  y  .
Solution:
The coefficients are 1, 4, 6, 4, 1 (from Pascal’s triangle).
The expansion is:
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 3x  y   1(3x)4  4(3x)3 y  6(3x)2 y 2  4(3x) y 3  y 4
 1 81x 4  4  27 x3 y  6  9 x 2 y 2  12 xy 3  y 4
 81x 4  108 x3 y  54 x 2 y 2  12 xy 3  y 4
Example 2:
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Find the first 4 terms in the expansion  2a  3b  .
Solution:
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First note that  2a  3b   (2a  (3b))10
The first 4 coefficients from Pascal’s triangle are
10
10
C0  1
C1  10
So
10
 2a  3b   (2a  (3b))10
C2  45
10
 1(2a)10  10(2a)9 (3b)  45(2a)8 ( 3b) 2  120(2a) 7 ( 3b)3
 1024a10  10  512a 9  3b  45  256a8  9b 2  120 128a 7  27b3
 1024a10  15360a 9b  103680a8b 2  414720a 7b3
C3  120 .
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Example 3:
1

Find the coefficient of x y in the expansion of  x  3 y 
2

Solution:
The value from Pascal’s triangle is 8C4  70.
The actual term is
4
1
1 
70   x  (3 y ) 4  70  x 4  81 y 4
16
2 
4
8
4
5670 4 4
x y
16
2835 4 4

x y
8

Revision Questions
1.
Find the expansion of (3x – y)5
2.
Find the coefficient of y2 in the expansion of (2y + 7)3.
3.
a) Show that (2  x)4  16  32 x  24 x 2  8 x3  x 4 .
b) Find the values of x for which  2  x   16  16 x  x 4 .
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4.
Find the non-zero value of b if the coefficient of x 2 in the expansion of  b  2 x  is equal to
6
the coefficient of x 5 in the expansion of  2  bx  .
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