Binomial Theorem - Haringeymath's Blog
Binomial Theorem
The Binomial Theorem is used to expand out brackets of the form [pic], where n is a whole number.
|n |[pic] |Coefficients |
|0 |[pic] |1 |
|1 |[pic] |1 1 |
|2 |[pic] |1 2 1 |
|3 |[pic] |1 3 3 1 |
|4 |[pic] |1 4 6 4 1 |
Note 1: The coefficients in these expansions form Pascal’s Triangle. These numbers can also be found using the [pic] button on a calculator. For example, the coefficients for the expansion of [pic]are:
[pic]
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, [pic]
Example 1:
Find the expansion of [pic].
Solution:
The coefficients are 1, 4, 6, 4, 1 (from Pascal’s triangle).
The expansion is:
[pic]
Example 2:
Find the first 4 terms in the expansion [pic].
Solution:
First note that [pic]
The first 4 coefficients from Pascal’s triangle are
[pic].
So
[pic]
Example 3:
Find the coefficient of [pic] in the expansion of [pic]
Solution:
The value from Pascal’s triangle is [pic]
The actual term is
[pic]
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 [pic].
b) Find the values of x for which [pic].
4. Find the non-zero value of b if the coefficient of [pic] in the expansion of [pic] is equal to the coefficient of [pic] in the expansion of [pic].
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