Friday 6 May 2016

Bionomial expansion

Binomial Theorem
For some basic values:


Binomial Expansion

For any power of n, the binomial (a + x) can be expanded


This is particularly useful when x is very much less than a so that the first few terms provide a good approximation of the value of the expression. There will always be n+1 terms and the general form is:

Some basic properties
Funda: There is one more term than the power of the
exponent, n. That is, there are terms in the expansion
of (a + b)n
.
Funda: In each term, the sum of the exponents is n,
the power to which the binomial is raised.
Funda: The exponents of a start with n, the power of
the binomial, and decrease to 0. The last term has no
factor of a. The first term has no factor of b, so
powers of b start with 0 and increase to n.
Funda: The coefficients start at 1 and increase
through certain values about “half”-way and then
decrease through these same values back to 1.
Funda: To find the remainder when (x + y)n
is divided
by x, find the remainder when yn
is divided by x.
Funda: (1+x)n ≅ 1 + nx,

When binomial expressions are expanded, is there any type of pattern developing which might help us expand more quickly?  Consider the following expansions:

What observations can we make in general about the expansion of (a + b)n ?

1. The expansion is a series (an adding of terms).
2.The number of terms in each expansion is one more than n.  (terms = n + 1)
3.The power of starts with an and decreases by one in each successive term ending with a0.  The power of bstarts with b0 and  increases by one in each successive term ending with bn.
4.The power of b is always one less than the "number" of the term.  The power of a is always n minus the power of b.
5. The sum of the exponents in each term adds up to n. 
6.The coefficients of the first and last terms are each one.
7. 
The coefficients of the middle terms form an interesting (but perhaps not easily recognized) pattern where each coefficient can be determined from the previous term.  The coefficient is the product of the previous term's coefficient and a's index, divided by the number of that previous term.
         Check it out:  
The second term's coefficient is determined by a4
The third term's coefficient is determined by 4a3b
To Get Coefficient
From the Previous Term:
(This pattern will eventually be expressed as a combination of the form n C k..)
8.Another famous pattern is also developing regarding the coefficients.  If the coefficients are "pulled off" of the terms and arranged, they form a triangle known as Pascal's triangle.  (The use of Pascal's triangle to determine coefficients can become tedious when the expansion is to a large power.)
1
1    1
1    2    1
1    3    3    1
1     4    6     4    1
(notice the symmetry of the triangle)
 
(The two outside edges of the triangle are comprised of ones.  The other terms are each the sum of the two terms immediately above them in the triangle.)

By pulling these observations together with some mathematical syntax, a theorem is formed relating to the expansion of binomial terms:
Binomial Theorem(or Binomial Expansion Theorem)
Most of the syntax used in this theorem should look familiar. The notation is just another way of writing a combination such as n C k  (read "n choose k").

(Ref from wild math) 
Our pattern to obtain the coefficient using the previous term (in observation #6), actually leads to the n C k used in the binomial theorem.
Here is the connection.  Using our coefficient pattern in a general setting, we get:

Let's examine the coefficient of the fourth term, the one in the box.
If we write a combination 
n C  using k = 3, (for the previous term), we see the connection:


The Binomial Theorem can also be written in its expanded form as:

Remember that 
 and that 
Examples using the Binomial Theorem:
1.  Expand  .
Let a = x, b = 2, = 5 and substitute.  (Do not substitute a value for k.)
    




Ref from (purplemaths,regentspre.org)

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