Tuesday, 24 January 2012

The Binomial Expansion

  • Edexcel, Module - C2, Chapter- Sequences and Series
  • AQA, Module - C2, Chapter - Sequences and Series
  • OCR, Module - C2, Chapter - Sequences and Series
 The Sequence and Series chapter in c2, is quite big, so I will divide it into 3 / 4 posts. a) Binomial Expansion 1 b) Binomial Expansion 2, c) Geometric Sequences 1 ,d) Geometric Sequences 2. 

The Binomial Expansion, is a theorem which allows us to expand (a + b)^n, where n is an integer. (a and b are just what i've used it, they can be any letters).

Why we need it ?

Let's say we have to expand the following terms :

(x + y)^1 = x + y
(x + y)^2 = x^2 + 2xy + y^2
(x + y)^3 = (x + y)(x + y)^2
               = (x + y)(x^2 + 2xy + y^2)
               = x^3 + 3x^2y + 3xy^2 + y^3

As we see, when increasing the power, (n) we get more and more terms, and it becomes more and more confusing, say we had to expand (x + y)^7, this would mean we would get 8 terms, after expanding the bracket 7 times, and then simplifying it. This would be a tedious and extremely long !

The Binomial theorem therefore allows us to expand the binomials ( two terms e.g. x + y or x + 3y or a - 3b). In each term of the expansion, there will be a coefficient of the term, these coefficients are either determined by Pascal's Triangle or Factorial Formula.

Let's see how this works, with the examples above... Let's expand (x + y)^3 using the theorem :

Binomial Theorem = To expand the binomial (x + y)^n, it can be written in the form :

This is the binomial coefficient, where n is the power and k is the increasing value till we reach n.

Let's use take our example using the theorem :

(x + y)^2 = (2,0) x^2 y^0 + (2,1) x^1 y^1 + (2,2) x^0 y^2

*The brackets are the binomial coefficients in blue , which are calculated using the formula :

* Notice the x terms start with the power n (in this case 2), and decrease by one power the next term, till the reach the power 0, simultaneously the y terms start with the power 0, and increase by one power the next term, till they reach the power n.

* We finish the binomial expansion when the binomial coefficient is (n,n) in this case (2,2), and when the x term has reached to the power 0 (which is 1), and the y term is to power n (in this case 2) . (Remember anything to the power 0 = 1 and anything to the power 1 = itself )

Let's simplify it down further, these are the binomial coefficients for each term :

! = Factorial (it calculates the product of all the positive integers less than or equal to some integer n)
(e.g. 4! = 4 x 3 x 2 x 1 = 24)

Note :  1! = 1 and  0! = 1

(2,0) = 2! / [ 0! (2 -0)! ]
         = 2 / [ 1 x 2 ]
         = 2 / 2
         = 1

(2,1) = 2! / [ 1! (2 -1)! ]

         = 2 / [ 1 x 1 ]
         = 2 /1
         = 2

(2,2) = 2! / [ 2! (2 -2)! ]

         = 2 / [ 2 x 1]
         = 2 / 2

So we have calculated the binomial coefficients for the terms, giving us :

(x + y)^2 = x^2 + 2xy + y^2

Pascal's Triangle ( Second way to find binomial coefficients - helps when n is small)

Another way to find the binomial cofficients, is the Pascal's Triangle. We start with n = 0, and start with number 1. Each row, will list the binomial cofficients in order of the expansion, each starting with 1 and ending with 1, depending on n.

To find the coefficient underneath, two gaps, we simply add the coefficients above it.

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

* The Triangle goes on as n increases, and the number of coefficients is always ( n + 1)
* The Triangle is symmetrical halfway, so we see that the first half of the coefficients of terms, is always the same as the second half
* Each row of the triangle starts and ends with 1 (meaning the first coefficient and last coefficient of any expansion is always 1)

Double check the expansion (x + y)^2 , where n = 2,
Check the row n = 2

The binomial coefficients should be 1, 2 and 1

1x^2 + 2xy + 1y^2

So we've seen how to expand expressions in the form (x + y)^n, and use two methods of working out the binomial coefficients. In the next post I will show you a couple of examples, and another expansion theorem, which allows us to expand algeabraic expressions in the form (1 + x)^ n.


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