## Binomial Expansions

topic explains a method of expanding binomial expressions such as (x + 3)7 without having to multiply out the brackets.

### The Binomial Coefficients

A coefficient is the number in an algebraic term. e.g. In 7x3 the coefficient is 7.

binomial coefficient is the coefficient of a term in a binomial expansion of the form (a + b)n

Investigation of the binomial expansions for different values of n reveals some interesting results:

 binomial binomial expansion binomial coefficients (a + b)0 1 1 (a + b)1 a + b 1   1 (a + b)2 a2 + 2ab + b2 1   2   1 (a + b)3 a3 + 3a2b + 3ab2 + b3 1   3   3   1 (a + b)4 a4 + 4a3b + 6a2b2 + 4ab3 + b4 1   4   6   4   1 (a + b)5 a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4+ b5 1   5   10   10   5   1

Note

• The triangle of binomial coefficients is known as Pascal's Triangle.
• Note that each value is the sum of the two above it.
• Notice also that there is always (n + 1) terms for a binomial to the n th power.
• The exponent of a decreases by 1 from left to right.
• The exponent of b increases by 1 from left to right.

The binomial coefficients can also be found from the combination formula:

For (a + b)4 the coefficients are 1, 4, 6, 4, 1 which is the same as .
These combinations can be looked up in the table of Binomial Coefficients.

All of this, leads to the Binomial Theorem

### The Binomial Theorem

This states that: It is true for all values of a and b and at this level we use it only for n being a positive integer. {1, 2, 3...}

Example = 1.32x5 + 5.16.3.x4 + 10.8.9.x3 + 10.4.27.x2 + 5.2.81. x + 1.243

= 32x5 + 240x4 + 720x3 + 1080x2 + 810x + 243

### The General Term

Often, especially when n is large a specific term is required. e.g the seventh term.

To do this the general term is used: Note that this is the (r + 1)th term not the r th term

Example

Find the 7th term of (x + 3)12 Much easier than expanding out the bracket 12 times!