Binomial Expansion

 

 

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Binomial Expansions

 

Factorials

 

Combination Symbol

 

Binomial Theorem

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Binomial Series

 

Algebra Main Page
 Binomial Theorem
for positive integral index
 
 
By observing Pascal's triangle and relating it to the combination symbol, it is realised that it can be re-written as:
 
n = 1                                   1C0   1C1
n = 2                                2C0   2C1   2C2 
n = 3                             3C0   3C1  3C2   3C3 
n = 4                          4C0   4C1  4C2   4C3   4C4
n = 5                       5C0    5C1  5C2   5C3   5C4  5C5 
 
Hence, the expansion of (1 + b)n is
     (1 + b)n = nC0 b0 + nC1 b1 + nC2 b2 + ... + bn
                  = 1 + nb + nC2 b2 + ... + bn
 
General Form of the Binomial Theorem
 
Not always would you see expressions in the form (1 + b)n, though most can be manipulated to that form. The more general form of the theorem is:
 
              (a + b)n = an + nC1 an-1b + nC2 an-2b2 + ... + bn
 
Observations:
 
1. The expansion is a finite series with (n + 1) terms
 
2. The expansion is in descending powers of a or ascending powers of b.
 
3. The powers of a and b always add up to n.
 
Expand (1 + 3x)5.
 
  Solution:
     (1 + 3x)5 = 1 + 5 (3x) + 5C2 (3x)2 + 5C3 (3x)3 + 5C4 (3x)4 + (3x)5 
                  = 1 + 15x + 90x2 + 270x3 + 405x4 + 243x5
 
Find the first three terms in the expansion
                   
   Solution:
     
                   =
                   = x7 - 7x4 + 21x + ...
 
Write down and simplify, in descending powers of x, the expansions of
           (3x + 1)4 and .
     Hence obtain the coefficient of x4 in the expansion of
            .
   Solution:
 
     (3x + 1)4 = (3x)4 + 4 (3x)3 + 4C2 (3x)2 + 4C3 (3x) + 1 
                  = 81x4 +108x3 +54x2 + 12x + 1
 
    
               
 
     Hence,
     
             
 
     The coefficient is 3727.
 
 
The next page will deal with more properties of the binomial theorem.