The Binomial Series
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You should know that (a + b)² = a² + 2ab + b² and you should be able to work out that (a + b)³ = a³ + 3a²b + 3b²a + b³ .
It should also be obvious to you that (a + b)¹ = a + b .

so (a + b)¹  =        a + b
(a + b)² =       a² + 2ab + b²
(a + b)³ =  a³ + 3a²b + 3b²a + b³

You should notice that the coefficients of a and b are:
     1   1
  1   2   1
1   3   3   1

If you continued expanding the brackets for higher powers, you would find that the sequence continues:
      1   4   6   4   1
   1   5  10  10  5   1
1   6  15  20  15  6   1
etc
This sequence is known as Pascal's triangle. Each of the numbers is found by adding together the two numbers directly above it.
So the 20 in the last line is found by adding together 10 and 10. The 10s in the line above are found by adding together 6 and 4.

So it is possible to expand (a + b) to any integral power using Pascal's triangle.

It is, of course, often impractical to write out Pascal's triangle every time, when all that we need to know are the entries on the
nth line. Clearly, the first number on the nth line is 1. The second number is n. The third number is:
n(n - 1)   .
 1 × 2

In general, the rth number in the nth line is:
    n!      (which is nCr on your calculator)
r! (n - r)!

Example:
Expand (2 + 4x)³
= 2³ + 3(2² × 4x) + 3(2 × 16x²) + 64x³
= 8 + 48x + 96x² + 64x³

© Matthew Pinkey

Other Notes in this Category

1. Algebraic Long Division
2. Functions
3. Indicies
4. Logarithms
5. Partial Fractions
6. Reduction to Linear Form
7. Sequences
8. Series
9. Set Theory
10. Simultaneous Equations
11. Surds
12. The Binomial Series

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