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].

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download