Binomial Theorem

The Binomial Theorem is used to expand out brackets of the form , where n is a whole number.

n Coefficients 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1

Note 1: The coefficients in these expansions form Pascal's Triangle. These numbers can also be found using the button on a calculator. For example, the coefficients for the expansion of are:

Note 2: As the power of a decreases by 1, the power of b increases by 1. In each term, when you add together the powers of a and b together you get n.

So,

Example 1: Find the expansion of .

Solution: The coefficients are 1, 4, 6, 4, 1 (from Pascal's triangle). The expansion is:

Example 2: Find the first 4 terms in the expansion .

Solution: First note that The first 4 coefficients from Pascal's triangle are . So Example 3: Find the coefficient of in the expansion of Solution: The value from Pascal's triangle is The actual term is

Revision Questions

1. Find the expansion of (3x - y)5

2. Find the coefficient of y2 in the expansion of (2y + 7)3.

3. a) Show that . b) Find the values of x for which .

4. Find the non-zero value of b if the coefficient of in the expansion of is equal to the coefficient of in the expansion of .