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Contents
Binomial Expansions
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Binomial Expansion
A binomial is an expression that
contains two terms. Examples include (a + b), (1 - a) and (k - h).
In algebraic manipulation, you
would have learnt the expansion of (1 + x)2, which is 1 + 2x + x2.
Using the same idea, the expansion of (1 + x)3 can be easily found:
(1 + x)3 = (1 + x)2(1 + x)
= (1 + 2x + x2) (1 + x)
= 1 + 2x + x2 + x + 2x2 + x3
= 1 + 3x + 3x2
+ x3
Similarly, we can use this method
to find (1 + x)4, (1 + x)5 and (1 + x)6. What
about, say, (1 + x)50 ?
Are we going to keep on expanding, probably taking a few days at it?
Pascal's Triangle
Look at the following expansions of
(1 + x)n:
(1 + x)0
=
1
(1 + x)1
=
1 + x
(1 + x)2
=
1 + 2x + x2
(1 + x)3
=
1 + 3x + 3x2 + x3
(1 + x)4
=
1 + 4x + 6x2 + 4x3 + x4
(1 + x)5
=
1 + 5x + 10x2 + 10x3 + 5x4 + x5
List the coefficients in a
triangular array:
n =
0
1
n =
1
1 1
n =
2
1 2 1
n =
3
1 3 3 1
n =
4
1 4 6 4
1
n =
5
1 5 10
10 5 1
This array is known as the Pascal's
Triangle.
 By
observation, it is noticed that each number in the array is the sum of the two
numbers above it. For example, in the last line, the third number is 10, which
is the sum of 4 and 6. This way, the binomial expansions of the following
expansions can be found.
There is a way of obtaining the
binomial coefficients directly, and this is by using the binomial theorem.
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